tilburg university irregular grid methods for pricing high

Tilburg University

Irregular grid methods for pricing high-dimensional American options

Berridge, S.J.

Publication date:2004

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Irregular Grid Methods for PricingHigh-Dimensional American Options

Irregular Grid Methods for PricingHigh-Dimensional American Options

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Universiteit van

Tilburg, op gezag van de rector magnificus, prof.dr. F.A. van der

Duyn Schouten, in het openbaar te verdedigen ten overstaan van

een door het college voor promoties aangewezen commissie in

de aula van de Universiteit op vrijdag 18 juni 2004 om 16.15 uur

door

STEFFAN JOHN BERRIDGE

geboren op 21 maart 1974 te Londen, Engeland.

PROMOTOR: PROF.DR. J.M. SCHUMACHER

to those with the courage to be irregular,

boe uiptf xip dbo xpsl pvu xibu uijt tbzt ...

Preface

This thesis consists of seven chapters, of which Chapters 2–4 are published in the

CentER Discussion Paper series as [8, 10, 11]. Chapters 2–4 and 6 were written in

cooperation with Hans Schumacher. Early versions of Chapter 2 are also published

as [7, 9].

vii

Acknowledgements

As my Tilburg era draws to a close I would like to thank those who have taught me

how use laserjet ink to turn paper into papers, and those who made four years in

Tilburg seem like one ride on a roller coaster.

The star of the official show was Hans, whose patience and encouragement

motivated me at every meeting. His guidance has contributed in no small way to

my inspiration for the contents of this thesis. Thank you Hans.

Thanks also go to the other members of the committee; Arun Bagchi, Gul

Gurkan, Jong-Shi Pang, Antoon Pelsser, Dolf Talman and Bas Werker; both for

reading the thesis and for providing valuable feedback.

In the last four years I have had contact with many people in the Faculty of

Economics and Business Administration, who have helped and guided me in many

ways. In particular I would like to thank the administrative staff at CentER and

the Department of Econometrics and Operations Research, who have always been

friendly and ready to help.

I have been motivated in my work through several other means. Participation

in the Mathematical Research Institute graduate programme took me to the heart

of current knowledge and research activities in mathematical finance, and I would

like to thank the lecturers and students for providing motivation and interesting dis-

cussions. Attendance of many conferences and workshops put me in contact with

important researchers and fellow students from other institutions. These valuable

experiences have inspired me and given me perspective in the academic world.

On the nonacademic side of life, there are many people I would like to thank

for making my time in Tilburg so enjoyable. The fellow inhabitants of Robert

Fruinstraat - Giorgio, Pedro, Zhang, Rossella and the mouse - and the inhabitants

of Mieredikhof - Charles, Vera, Jeroen, Pierre-Carl, Sabine and the mouse - have

ix

always provided great company, interesting conversation and a range of culinary

experiences. Thanks to those who helped set up the Mieredikhof Market Gardening

enterprise, to the neighbours who seldom complained about this and other activi-

ties, and to one neighbour for so thoughtfully and frequently stating his outspoken

perceptions on the odour of barbecues.

Of course any discussion about life in Tilburg cannot omit mention of the social

genius of the Tilburg University International Club (Tunic) of which Aldo and

Sandra have recently been the backbone, organising activities from pot luck dinners

to kart racing. Then there’s the people who selflessly attended the bierproeverij at

Kandinsky on Sunday afternoons - these are too many to mention, but perhaps I

should thank the waiters for always meticulously stating the alcohol percentage of

the beer and its origin, and for never tiring of the question ”Is that in Belgium?”.

I would also like to extend thanks the owner of the Villa for providing a meeting

point for social activities and a great venue for celebrating various occasions in a

modest but enjoyable fashion, and to the creator of the inspiring statue on the right

inside the church on Heuvelplein.

Special thanks to those who helped in setting up the Graduate Students’ So-

ciety. In particular the other founding board members Pierre-Carl, Rossella, Ania

and Man-Wai. Also to the other students who selflessly gave their time and en-

ergy to be part of the society, to Niels van de Ven from the university bureau who

provided much valuable advice on setting up the society, and to the faculty for

providing funding for this initiative. My life would also not have been complete

without GraDUAL magazine, with its dazzling feature articles and revealing stu-

dent mini-autobiographies, so thanks to the people who made it possible (however

”GraDUAL-ly” it comes out.)

On the sporting side, thanks to the guys and occasional girl that participated in

the soccer games on Friday afternoons, and to the main organisers Bram, Jeroen

and Mark-Jan. Thanks also to the people who encouraged me with running, mainly

spearheaded by Bertrand Melenberg, and to those who joined on countless occa-

sions running in the woods and along the picturesque Wilhelmina canal. Special

thanks to Pierre-Carl for being far more attractive to large dogs than I was. On

the culinary side, thanks to Tam Food for providing a great selection of inexpen-

sive Chinese condiments. And how could I forget I Pin Ke, Bali and Minos Pallas

which formed the hard core of good quality but reasonably-priced restaurants in

Tilburg.

My Tilburg experience has also been enriched by participating in the university

students’ choir Contrast, conducted by Peter van Aerts. Aside from some spectac-

ular performances, including Carmina Burana and Mozart’s Requiem, I have also

enjoyed having a few quiet pints with fellow choir members at the Korenbloem on

Monday evenings, and attending rehearsal weekends. Thanks Contrasters.

Finally, I would like to thank Vera, whose honesty, kindness and determination

have added a new dimension to my life.

STEFFAN BERRIDGE

APRIL 8TH 2004, TILBURG

Contents

1 Introduction 1

2 A Method Using Matrix Roots 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 State space discretisation . . . . . . . . . . . . . . . . . . 15

2.3.2 Approximating the partial differential operator . . . . . . 16

2.3.3 Time discretisation . . . . . . . . . . . . . . . . . . . . . 17

2.3.4 Randomisation . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.5 Summary of procedure . . . . . . . . . . . . . . . . . . . 21

2.4 Experimental setup and details . . . . . . . . . . . . . . . . . . . 21

2.4.1 Specification of dynamics . . . . . . . . . . . . . . . . . 21

2.4.2 Elimination of drift . . . . . . . . . . . . . . . . . . . . . 22

2.4.3 Grid specification . . . . . . . . . . . . . . . . . . . . . . 22

2.4.4 Reuse of roots for similar processes . . . . . . . . . . . . 23

2.4.5 Low-biased estimate . . . . . . . . . . . . . . . . . . . . 26

2.4.6 High-biased estimate . . . . . . . . . . . . . . . . . . . . 26

2.4.7 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 A Method Using Local Quadratic Approximations 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 The pricing problem . . . . . . . . . . . . . . . . . . . . . . . . 42

xiii

3.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.3 Nearest neighbours . . . . . . . . . . . . . . . . . . . . . 44

3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Approximating the differential operator . . . . . . . . . . 44

3.3.2 Weighted least squares . . . . . . . . . . . . . . . . . . . 47

3.3.3 Time stepping . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Application to regular grid . . . . . . . . . . . . . . . . . . . . . 48

3.4.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.2 Two dimensions . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 Grids on the unit cube in R2 . . . . . . . . . . . . . . . . 54

3.5.2 Normally distributed grids in R2 . . . . . . . . . . . . . . 57

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 A Method Using Local Consistency Conditions 614.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.1 The market . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 Irregular grid . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.2 Approximation of Markov chain . . . . . . . . . . . . . . 67

4.3.3 Approximation of infinitesimal generator . . . . . . . . . 68

4.3.4 Time stepping . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.5 Summary of the algorithm . . . . . . . . . . . . . . . . . 70

4.4 Fine tuning and extensions . . . . . . . . . . . . . . . . . . . . . 70

4.4.1 Grid specification . . . . . . . . . . . . . . . . . . . . . . 71

4.4.2 Boundary region and boundary conditions . . . . . . . . . 72

4.4.3 Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4.4 Control variates and Richardson extrapolation . . . . . . . 74

4.4.5 Matrix reuse . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4.6 Grid expansion . . . . . . . . . . . . . . . . . . . . . . . 76

xiv

4.4.7 Partially absorbing boundaries . . . . . . . . . . . . . . . 77

4.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5.1 Geometric average options . . . . . . . . . . . . . . . . . 77

4.5.2 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5.3 Experimental details . . . . . . . . . . . . . . . . . . . . 78

4.5.4 Experimental results . . . . . . . . . . . . . . . . . . . . 79

4.5.5 Error behaviour . . . . . . . . . . . . . . . . . . . . . . . 80

4.5.6 Timings . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5.7 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 A Method Using Interpolation 95

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Bermudan swaption pricing in the LIBOR

market model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.1 Setup of the LMM . . . . . . . . . . . . . . . . . . . . . 98

5.2.2 Swaption pricing . . . . . . . . . . . . . . . . . . . . . . 99

5.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.1 Framework and assumptions . . . . . . . . . . . . . . . . 100

5.3.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3.3 Approximating the quadrature operator . . . . . . . . . . 107

5.3.4 The interpolation operator IX . . . . . . . . . . . . . . . 107

5.3.5 Variance reduction . . . . . . . . . . . . . . . . . . . . . 109

5.3.6 Multiple grids . . . . . . . . . . . . . . . . . . . . . . . . 111

5.3.7 Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4.1 Bermudan geometric average options . . . . . . . . . . . 112

5.4.2 Bermudan swaptions . . . . . . . . . . . . . . . . . . . . 113

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6 Convergence Results 125

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.1 Stopping time formulation . . . . . . . . . . . . . . . . . 127

6.2.2 Variational inequality formulation . . . . . . . . . . . . . 128

xv

6.2.3 General variational inequality formulation . . . . . . . . . 129

6.3 Solution framework . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.3.1 Discretisation of space and time . . . . . . . . . . . . . . 133

6.3.2 Approximation of constraints and operators . . . . . . . . 139

6.4 Convergence of the discretised problems . . . . . . . . . . . . . . 142

6.4.1 Stability under M -matrix assumption . . . . . . . . . . . 142

6.4.2 Stability and convergence under Garding inequality

assumption . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.4.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . 154

6.5 Approximating the bilinear form . . . . . . . . . . . . . . . . . . 154

6.5.1 Local consistency method . . . . . . . . . . . . . . . . . 154

6.5.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 156

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7 Conclusions and Future Research 167

A Software 171

xvi

List of Figures

1.1 Aspects of a stochastic model. . . . . . . . . . . . . . . . . . . 2

1.2 Grid and time-state domain of approximating continuous-time

Markov chain . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Random grid valuation of an American put option on a single asset 19

2.4.1 Example of normal QMC grid in 2 dimensions with 500 points. . 24

2.5.1 Average QMC grid valuation over 50 normal grids, explicit . . . 31

2.5.2 Average QMC grid valuation over 50 normal grids, Crank-Nicolson 32

2.5.3 Average QMC grid valuation over 50 normal grids, implicit . . . 33

3.3.1 Mapping of eigenvalues from generator matrix A to time step-

ping matrix M . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.1 Points and eigenvalues of A for regular grid with 6 neighbours . 55

3.5.2 Points and eigenvalues of A for regular grid with 9 neighbours . 55

3.5.3 Points and eigenvalues of A for triangular grid with 6 neighbours 55

3.5.4 Points and eigenvalues of A for hexagonal grid with 6 neighbours 56

3.5.5 Points and eigenvalues of A for uniform pseudo-random grid

with 6 neighbours . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5.6 Points and eigenvalues of A for Sobol’ grid with 6 neighbours . 56

3.5.7 Points and eigenvalues of A for low distortion grid with 6 neigh-

bours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.8 Points and eigenvalues of A for low distortion normal grid with

r = ∞ and 6 neighbours . . . . . . . . . . . . . . . . . . . . . 58


r = 3.0 and 6 neighbours . . . . . . . . . . . . . . . . . . . . . 59

xvii


r = 2.5 and 6 neighbours . . . . . . . . . . . . . . . . . . . . . 59


r = 2.0 and 6 neighbours . . . . . . . . . . . . . . . . . . . . . 59

4.3.1 Grids with 500 points adapted to the normal density. . . . . . . . 66

4.4.1 Interior and boundary points on a normal low distortion grid . . 72

4.4.2 Naive prediction for boundary behaviour . . . . . . . . . . . . . 74

4.5.1 Bermudan pricing results for normal Sobol’ grids . . . . . . . . 81

4.5.2 American pricing results for normal Sobol’ grids . . . . . . . . 87

4.5.3 Bermudan pricing results for low distortion grids . . . . . . . . 87

4.5.4 American pricing results for low distortion grids . . . . . . . . . 88

4.5.5 Log of absolute errors for geometric average options for normal

Sobol’ grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.5.6 Log of absolute errors for geometric average options for low dis-

tortion grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.5.7 Observed boundary behaviour for normal Sobol’ grids . . . . . . 92

4.5.8 Observed boundary behaviour for low distortion grids . . . . . . 93

5.4.1 Bermudan geometric average option estimates . . . . . . . . . . 117

5.4.2 Timings for Bermudan geometric average option estimates . . . 118

5.4.3 Bermudan geometric average option estimates for nearest neigh-

bour interpolation and integrating the continuation value . . . . 119

5.4.4 Bermudan geometric average option estimates for nearest neigh-

bour interpolation and integrating the early exercise premium . . 120

5.4.5 Swaption estimates for contracts I–IV and volatility scenario C . 121

5.4.6 Swaption estimates for contracts I–IV and volatility scenario D . 122

5.4.7 Timings for Bermudan swaption estimates . . . . . . . . . . . . 123

6.3.1 Voronoi cells of a grid X containing 20 points . . . . . . . . . . 135

6.5.1 Maximum eigenvalues of (6.5.5) for inverse normal regular and

Sobol’ grids in one dimension . . . . . . . . . . . . . . . . . . 158

6.5.2 Maximum eigenvalues of (6.5.5) for inverse normal regular grids

in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 159

xviii

6.5.3 Maximum eigenvalues of (6.5.5) for normal Sobol’ and low dis-

tortion grids in two dimensions . . . . . . . . . . . . . . . . . . 160


tortion grids in three dimensions . . . . . . . . . . . . . . . . . 160


tortion grids in four dimensions . . . . . . . . . . . . . . . . . . 161


tortion grids in five dimensions . . . . . . . . . . . . . . . . . . 161


tortion grids in six dimensions . . . . . . . . . . . . . . . . . . 162


tortion grids in seven dimensions . . . . . . . . . . . . . . . . . 162


tortion grids in eight dimensions . . . . . . . . . . . . . . . . . 163

6.5.10 Maximum eigenvalues of (6.5.5) for low distortion grids in 2–8

dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.5.11 Plot of the first stability quantity maxi |vin|

2n for dimensions 1, 2

and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.5.12 Plot of the second stability quantity∑

i |vi+1n − vi

n|2n for dimen-

sions 1, 2 and 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.5.13 Plot of the third stability quantity δt∑

i

(

θ‖vi+1n ‖2

n + (1 − θ)‖vin‖

2n

)

for dimensions 1, 2 and 5 . . . . . . . . . . . . . . . . . . . . . 165

xix

List of Tables

2.5.1 Comparison of Bermudan price estimates with ten exercise op-

portunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5.2 Comparison of American price estimates . . . . . . . . . . . . . 35

2.5.3 Timings and storage requirements for irregular grid method . . . 37

3.4.1 Assignment of indices of v to neighbours. . . . . . . . . . . . . 51

3.4.2 Weights for α00, for finite difference and irregular grid, respectively. 52






4.5.1 Benchmark results for geometric average options in dimensions

1–10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.5.2 Results for Bermudan geometric average put options using nor-

mal Sobol’ grids . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5.3 Results for American geometric average put options on normal

Sobol’ grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5.4 Results for European geometric average put options on normal

Sobol’ grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5.5 Results for Bermudan geometric average put options on normal

Sobol’ grids with control variate . . . . . . . . . . . . . . . . . 83

4.5.6 Results for American geometric average put options on normal

Sobol’ grids with control variate . . . . . . . . . . . . . . . . . 84

xxi

4.5.7 Results for Bermudan geometric average put options using low

distortion grids . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5.8 Results for American geometric average put options using low


4.5.9 Results for European geometric average put options using low


4.5.10 Results for Bermudan geometric average put options using low

distortion grids, with control variate . . . . . . . . . . . . . . . 86

4.5.11 Results for American geometric average put options using low

distortion grids, with control variate . . . . . . . . . . . . . . . 86

4.5.12 Regression coefficients for the error behaviour on normal Sobol’

grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.5.13 Regression coefficients for the error behaviour on low distortion

grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4.1 Parameters of the Bermudan swap option contracts . . . . . . . 114

xxii

Chapter 1

Introduction

The field of option pricing, and more generally mathematical finance, has rep-

resented one of the great triumphs of interdisciplinary research in the twentieth

century. Louis Bachelier, aptly named the founder of mathematical finance, set

the stage for its study in 1900 with his doctoral thesis “Theorie de la Speculation”

which proposed continuous-time stochastic processes, including Brownian motion,

as a basis for the analysis of financial markets. Seventy-three years later Fischer

Black, Myron Scholes and Robert Merton revolutionised the study of option pric-

ing by devising closed-form expressions for European option prices based on a

model in which the underling asset follows a geometric Brownian motion. In the

same year the Chicago Board Options Exchange was founded and the rest, as they

say, is history.

The study of option pricing centres on the modelling of financial market pro-

cesses as simplified stochastic processes, the statistical analysis connecting the

model with relevant data and the mathematical analysis of such stochastic mod-

els. The statistical analysis may involve the estimation of model parameters or the

extraction of market information related to the model. The mathematical analysis

typically starts with a particular stochastic model and attempts to draw relevant

conclusions, for example regarding option prices, assuming that the model is cor-

rect. This field of study thus draws together, at least in the first instance, researchers

from finance, econometrics, statistics and mathematics.

This thesis focuses entirely on the mathematical aspect of option pricing, and

completely ignores the statistical aspect. Given a stochastic model for a number

1

2 Chapter 1. Introduction

Stochastic Model

Real-world Problem

Statistical Analysis Mathematical Analysis

Figure 1.1: Aspects of a stochastic model.

of possibly interdependent financial market processes, the question of interest is

how one can efficiently determine a no-arbitrage price for a derivative based on

these processes. In particular we ask this question for American options written on

multidimensional processes which follow correlated geometric Brownian motions.

Thus the empirical question of whether the models studied are realistic is not

tackled. For the purposes of the research however, the models are believed to con-

stitute an adequate representation of reality for many processes found in financial

markets. On the other hand, it is an uncontestable fact that many superior models

have been developed for various financial processes over the past three decades.

Such models have not been considered here because simpler models provide a suf-

ficiently rich testing ground on which to realise the main aims of the research.

As the title makes clear, this thesis tackles the problem of pricing American

options when the number of underlying variables (the dimension) is large. The

closely related problem of pricing of Bermudan options, where the number of ex-

ercise opportunities is finite, is included in the scope of the research. For “large”

one may read “at least three”, since this is the dimension in which classical so-

lution methods, in particular those based on regular grid discretisations, become

cumbersome. The thesis further focuses on methods which use an irregular grid as

a basis for calculations. The use of an irregular grid allows one to devise methods

which are based on a tractable number of points, and which allow some freedom in

placing more points in areas where the behaviour has more effect on the required

solution.

The combination of the early exercise feature, offered by both American and

Bermudan options, with the problem of high-dimensionality presents a consider-

Chapter 1. Introduction 3

able challenge for numerical analysis both in terms of the pricing problem and

the associated hedging problem. The early exercise feature itself does not present

a great challenge for one-dimensional problems; neither does high-dimensionality

present a great challenge when pricing options without early exercise features; such

problems can be solved relatively quickly using Monte Carlo or quasi-Monte Carlo

integration methods. In combining these two complications however, one appears

to require considerable computational resources.

An important question is whether the minimum amount of computational re-

sources required to solve the problem to within a certain accuracy must increase

exponentially with dimension; that is, are we facing a curse of dimensionality in

the sense of approximating the solution? This question is explored in a related

context by Rust [65], who shows that one can break the curse of dimensionality for

certain classes of Markovian decision problems. The regularity assumptions are

too strict for the types of option pricing problems considered in this thesis, and the

question thus remains open for the latter.

The first widely-used numerical methods for pricing American options were

those of Brennan and Schwartz [16] and Cox et al. [24], the former being an adapta-

tion of the explicit finite difference method used for pricing European options, and

the latter a binomial tree method. Later Wilmott et al. [74] showed how implicit

finite difference methods for solving partial differential equations (PDEs) could

be used for pricing American options by solving a linear complementarity problem

(LCP) at each step. They used the projected successive overrelaxation (PSOR) me-

thod of Cryer [25] to solve the LCPs; more recently Huang and Pang [40] provide

a review of state-of-the-art numerical methods for solving LCPs. Although these

methods were all successful for solving one-dimensional problems, their reliance

on regular grid discretisations rendered them unsuitable for high dimensions. Other

relevant work included the analytic valuations of Geske and Johnson [32] and a

number of methods involving approximations of the exercise boundary.

The first author to break ranks on the prevailing view that simulation tech-

niques could not be used for pricing American options was Tilley [70], who used

state space aggregation as a means for specifying the continuation values. This

had important consequences for high-dimensional problems because Monte Carlo

methods were at least able to make computations feasible in high dimensions, and

were even able to break the curse of dimensionality for certain problems. Two


years later, Barraquand and Martineau [4] presented similar methods specifically

aimed at the pricing of high-dimensional American securities.

The use of simulation for valuing American options was further developed by

Broadie and Glasserman [18], who used a simulated tree structure to calculate con-

fidence intervals for option values. The method is suitable for high-dimensional

problems, although it suffers exponential computational complexity in the number

of time steps. This method was extended by the same authors [19] using a stochas-

tic recombining mesh which did not suffer the curse of dimensionality, at least in a

computational sense.

Carriere [20] made an important development by showing that path simulations

of the underlying process could be used simultaneously to approximate the optimal

stopping time and to provide price estimates. This was done by using nonparamet-

ric techniques to estimate the continuation value, and using the implied stopping

rule to determine the average value realised along the simulated paths. Longstaff

and Schwartz [50] later showed this to be a feasible method for high-dimensional

problems using least squares regression in place of nonparametrics to estimate the

continuation value.

In related work, Tsitsiklis and Van Roy [71] use projection onto a set of “fea-

tures” to perform value iteration. The projection used here is in principle no differ-

ent to that performed by Longstaff and Schwartz, where the features are the basis

functions used for the regression. The key difference between the two methods

is in the use of information from the paths; Tsitsiklis and Van Roy perform value

iteration using information only from the function approximation made for the pre-

vious time step, whereas Longstaff and Schwartz use the implied stopping rule to

determine the realised value along each path. One can see this difference from

another angle — namely the difference in the use of the functional approximation.

Tsitsiklis and Van Roy use functional approximation directly to approximate the

value function; Longstaff and Schwartz use functional approximation to specify a

(hopefully near-optimal) stopping rule. This difference seems to weigh in favour

of Longstaff and Schwartz in terms of accuracy; suggesting that the value of an

American option is not very sensitive to the stopping rule used in the valuation.

Tsitsiklis and Van Roy prove convergence of their method in their paper, and a

proof for the Longstaff and Schwartz method is undertaken by Clement et al. [22].

The question about the rate of convergence has been formally answered only with


respect to the number of paths in the Longstaff and Schwartz method, where the

dependence is n−1/2; more difficult is to determine the rate of convergence with

respect to the number of basis functions. Given the use of functional approximation

methods, which are known to suffer a curse of dimensionality, one may fear the

worst. Recent analysis has focused on how the minimum number of paths required

for convergence depends on the number of basis functions. Stentoft [69] lays a

theoretical basis for requiring a cubic number of paths on an average case basis;

Glasserman and Yu [34] prove that the requirement on a worst case basis is at least

exponential.

This thesis contributes to the above literature by exploring new methods for

the valuation of high-dimensional American options. The methods presented are

different to those discussed above, except of course for their main aim which is to

specify computationally efficient methods. Rather than using path simulations, we

rather use an irregular grid as the means for storing information about the value

function. In Chapters 2–4 the grid is assumed to be constant over time, thus consti-

tuting a method of lines approach when the problem is seen from the PDE point of

view, or an approximating Markov chain approach when seen from the stochastic

differential equation (SDE) point of view. A proof of convergence for the methods

of Chapters 2–4 is presented in Chapter 6. The method presented in Chapter 5

differs fundamentally from the others in that the irregular grid is allowed to change

over time, demanding a different approach for the use of information from previous

time steps. A separate proof of convergence is provided for the method presented

in that chapter.

The use of an irregular grid is inspired by the use of similar methods for ap-

proximating integrals. The value of a European option is essentially the discounted

integral of the payoff function with respect to the density of the process at ex-

piry. This can be evaluated at least moderately efficiently using Monte Carlo (MC)

methods, and in many cases more efficiently using quasi-Monte Carlo (QMC), or

other low discrepancy- or low distortion-based methods. The use of simulation

methods for pricing high-dimensional European options is treated in Boyle et al.

[13]; another useful reference for MC integration is Evans and Swartz [28] and for

QMC integration Niederreiter [58]. The historical succession of methods contains

an interesting twist: firstly the deficiency of deterministic integration methods in

high dimensions led to the development of MC methods, which are randomised;


secondly the slow rate of convergence observed for MC methods led to the devel-

opment of the faster QMC methods, which are again deterministic. The question

we hope to answer in this thesis is whether the success of MC and QMC meth-

ods can be combined with traditional PDE solution methods to form a framework

for the numerical solution of optimal stopping problems, including the American

option pricing problem.

There are two main reasons to believe that PDE methods may be preferable

to MC methods for American option pricing. Firstly, PDE methods typically ad-

mit Taylor series error analyses for European problems, whereas simulation-based

methods admit less optimistic probabilistic error analyses; in practise one indeed

observes faster convergence rates for PDE methods. Secondly, the number of tun-

ing parameters that must be used in PDE methods is much smaller than that re-

quired for the type of simulation-based techniques that have been suggested for

American option pricing; for the latter one often faces the problem of having to

choose a suitable set of basis functions, for which choice one must have sufficient

a priori knowledge about the shape of the value function.

Like its European counterpart, the value of an American option is also a dis-

counted integral; the integral in the latter case is rather performed with respect to

the (unknown) optimal stopping boundary. Thus, for an American option one not

only requires some representation of the process density at expiry, but also of its

behaviour at intermediate times. The main challenge faced is thus the representa-

tion of the continuous process in a discrete state context, whether that be done in a

continuous or discrete time setting. In the case of continuous time one works on a

domain similar to that shown in Figure 1.2. In this case one requires an infinitesi-

mal generator matrix to provide a representation of the original process, and in dis-

crete time one requires a Markov transition matrix. In Chapter 2 the infinitesimal

generator matrix is constructed by taking the logarithm of a matrix corresponding

to transition probabilities at expiry; in Chapter 3, local quadratic approximations

for the value function are used to build the generator matrix, and in Chapter 4 local

consistency conditions similar to those of Kushner and Dupuis [46] are used.

One of the most difficult issues arising in irregular grid schemes turns out to be

stability. Indeed one may specify transition probabilities using a number of seem-

ingly consistent methods. The stability of such schemes is usually related to the

eigenvalues of the generator or transition matrix, and is thus difficult to guarantee


−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

x1

x 2

(a) Grid

00.5

1−20

2

−3

−2

−1

0

1

2

3

x1 t

x 2

(b) Time-state domain

Figure 1.2: Example of 50-point grid and corresponding time-state domain of ap-

proximating continuous-time Markov chain for methods of Chapters 2–4.

in many circumstances. Chapter 3 provides a case in point for this issue, where

the connection between the weights and the eigenvalues is difficult to ascertain.

Finding a way to force the eigenvalues into the stable region is thus very difficult,

although experiments suggest that stability may be induced by considering rela-

tively “smooth” grid specifications or by manipulating the neighbour configuration

so that neighbours are distributed about each point more uniformly in an angular

sense.

Chapters 2 and 4 admit simpler stability analyses. In the former, one obtains an

infinitesimal generator by taking the logarithm of a nonsingular Markov transition

matrix having eigenvalues in the unit interval, thus restricting the eigenvalues to

the negative half line. In the latter, the infinitesimal generator is specified using

local consistency conditions in combination with sign restrictions on the elements;

this leads to a diagonally dominant matrix with negative diagonal thus restricting

the eigenvalues to lie in the left half plane through Gershgorin’s disk theorem.

A closely related strand of literature to Chapters 2–4 is the work on sparse grids

initiated by Zenger [75], and continued by several German and Dutch researchers.

Sparse grids attempt to deal in particular with the solution of high-dimensional

PDE problems. The sparse grid approach differs in two important ways from the

one taken here; first it assumes a rectilinear domain for the problem and second

the sparse grid structure does not admit a simple stochastic interpretation. On

the other hand, the irregular grid methods considered in this thesis are meant in


principle to form a basis for the approximation of unbounded stochastic processes,

and the specification of transition intensities can be directly interpreted in terms

of the local behaviour of the original stochastic process. One may also view the

methods of Chapters 2–4 in a PDE context, facilitating the construction of implicit

time stepping schemes.

Another important related work is that of Kushner and Dupuis [46], who for-

mulate approximating Markov chain methods for solving optimal investment con-

sumption problems. Although the problems they consider are mostly of low di-

mension, their ideas of local consistency extend naturally to higher dimensions.

Their ideas have been used in Chapter 4 as a motivation for specifying transition

intensities and probabilities in approximating Markov chains on irregular grids.

Kushner and Dupuis do in fact present methods for approximating Markov chain

dynamics on certain nonstandard grids, but they do not consider irregular grids as

used in this thesis.

A proof of convergence related to the methods of Chapters 2–4 is presented

in Chapter 6. The proof makes use of the variational inequality formulation for

American option pricing, and follows the framework developed by Glowinski et

al. [35], and the references therein, in proving convergence of numerical schemes

for solving variational inequalities. The Glowinski et al. framework constitutes a

powerful tool due to its abstractness; Jaillet et al. [42] and Zhang [76] have previ-

ously used these methods for proving convergence of one-dimensional American

option pricing algorithms.

Chapter 5 presents a method which is fundamentally different from those found

in the other chapters. The essential difference is that a different grid is permitted at

each time step, thus allowing the grids to be more adapted to the process. Value it-

eration is performed, with information from the previous time step being conveyed

through appropriate interpolation and quadrature operators. The interpolation op-

erator extends information contained at grid points to define a value function on the

entire state space, and the quadrature operator uses this interpolated value function

as a basis for finding the expected value of continuation. This method is closely

linked to the literature on mesh-free methods which studies interpolation methods

that can be applied to irregular grids. Recent work by Levin [47] and Wendland

[72] shows that moving least squares methods can produce interpolants which are

accurate to high degrees, and work by Maz’ya and Schmidt [53] and Fasshauer [29]


hints at how one may obtain such interpolants in reasonable computation times.

These latter methods have not been employed in the current work, but they show

how the complexity may be reduced. Nearest neighbour interpolation is found to

be much faster than moving least squares, and gives reasonable pricing results with

the use of an inner control variate, i.e. the application of a control variate at each

time step.

Chapter 2

A Method Using Matrix Roots

2.1 Introduction

The pricing of American options has always required numerical solution methods;

in high-dimensional cases even the most sophisticated methods have difficulty in

providing accurate solutions. Given the practical importance of such cases, it is

of considerable interest to develop solution methods which are reliable and which

provide accompanying exercise and hedging strategies.

Barraquand and Martineau [4] are perhaps the first to consider pricing high-

dimensional American options specifically, proposing an algorithm based on the

aggregation of paths with respect to the intrinsic value. The method is difficult to

analyse and has a possible lack of convergence; Boyle et al. [12] demonstrate this

and propose a modification of the algorithm which leads to a low-biased estimator.

Broadie and Glasserman [18] use a stochastic tree algorithm to give both a

low-biased and a high-biased estimator of the price, both asymptotically unbiased.

They also argue that there exists no nontrivial unbiased estimator for the price.

Their method requires an exponentially increasing amount of work in the number

of exercise opportunities. In subsequent work [19] they present a related method

based on a stochastic mesh which does not suffer from this problem, although this

method has been found to be slow by several authors and to have a large finite-

sample bias (see e.g. Fu et al. [31]).

The least squares Monte Carlo (LSM) method presented by Longstaff and

Schwartz [50] attempts to approximate the price of an American option using

11

12 Chapter 2. Method Using Matrix Roots

cross-sectional information from simulated paths. The optimal exercise strategy

is successively approximated backwards in time on the paths by comparing the in-

trinsic values to the continuation values projected onto a number of basis functions

over the states. Experimental success is reported for the LSM method, although in

high dimensions the basis functions must be chosen carefully. Recently Clement

et al. [22] and Stentoft [69] independently provide proofs of convergence for the

LSM method, showing that the convergence rate is n−1/2 in the number of paths

used. The convergence behaviour in the number of basis functions however has not

been determined. Stentoft [69] and Glasserman and Yu [34] establish relationships

between the paths and number of basis functions which are necessary for conver-

gence; Stentoft finds that the number of paths should be greater than cubic in the

number of basis functions to achieve convergence in probability, while Glasserman

and Yu find the relationship should be exponential in the square for convergence

on a worst case basis. Stentoft [68] and Moreno and Navas [54] test the LSM al-

gorithm numerically. Stentoft suggests that basis functions up to order three are

sufficient in five dimensions for arithmetic and geometric average options, but not

for minimum or maximum options. Moreno and Navas find that the method is

sensitive to the choice of basis functions in five dimensions.

Tsitsiklis and Van Roy [71] propose a method similar to LSM where approx-

imate value functions are projected onto an orthogonal set of basis functions, the

orthogonality being with respect to a suitably chosen inner product which in gen-

eral changes between time periods. They provide a proof of convergence but no

empirical results. The method differs from LSM in that the projection is used to

determine an approximate value function rather than an exercise rule.

Boyle et al. [14] recently extended the stochastic mesh method of Broadie and

Glasserman [19] with their low discrepancy mesh (LDM) method. This involves

generating a set of low discrepancy interconnected paths and using a dynamic pro-

gramming approach to find prices on the mesh.

An interesting alternative approach is proposed independently by Rogers [64]

and Haugh and Kogan [37] and later developed by Jamshidian [44] and Kolodko

and Schoenmakers [45]. They use a dual formulation of the problem in which a

minimisation is performed over martingales. The method is sensitive to the choice

of basis martingales chosen to perform the minimisation, and so requires the basis

to be well-chosen in order to give an accurate solution. The method gives a high-

2.2. Formulation 13

biased estimator.

In this chapter, we propose a new approach to solving the American option

pricing problem inspired by the success of numerical integration in high dimen-

sions and related to the method of lines for solving partial differential equations

(PDEs).

We first perform a discretisation of the state space using quasi-Monte Carlo

(QMC) points, the points being taken with respect to an importance sampling dis-

tribution related to the transition density of the process at expiry. We then pro-

pose an approximation to the partial differential operator on this grid by taking the

logarithm of a transition probability matrix P (T−t) which approximates the joint

density of the underlyings at the expiry of the option, T − t. This approximation is

then used to formulate linear complementarity problems (LCPs) at successive time

points, working back from the option expiry.

We propose an implementation of this method in which the matrix logarithm

of P (T−t) does not need to be calculated explicitly, but instead a root of the matrix

can be calculated. The root operation is cheaper than the logarithm, although the

logarithm allows variation of the time step without recalculation. The computa-

tional elements of the method are thus the QMC trials, the generation of the matrix

P (T−t), the matrix root and solving an LCP at each time step. For approximating

the European option price this method amounts to performing a numerical integra-

tion with importance sampling, which is known to be an efficient method in high

dimensions as long as the importance sampling distribution is chosen appropriately.

The remainder of this chapter is organised as follows. In Section 2.2 we present

a mathematical formulation of the problems to be solved numerically and in Sec-

tion 2.3 we show how an irregular grid method can be used to solve the problem.

We then present the experimental setup in Section 2.4, results in Section 2.5 and

concluding remarks in Section 2.6.

2.2 Formulation

We consider a complete and arbitrage-free market described by state variable X(s)

∈ Rd for s ∈ [t, T ] which follows a Markov diffusion process

dX(s) = µ(X(s), s)ds+ σ(X(s), s)dW (s) (2.2.1)


with initial condition X(t) = xt, and a derivative product on X(s) with intrinsic

value ψ(X(s), s) at time s and value V (s) = v(X(s), s) for some pricing function

v(x, s). The process V (s) satisfies

dV (s) = µV (X(s), s)ds + σV (X(s), s)dW (s) (2.2.2)

where µV and σV can be expressed in terms of µ and σ by means of Ito’s lemma.

The terminal value is given by v(·, T ) = ψ(·, T ).

In such a market there exists a unique equivalent martingale measure under

which all price processes are martingales. The risk-neutral process in this case is

given by

dX(s) = µRN (X(s), s)ds + σ(X(s), s)dW (s) (2.2.3)

where µRN is the risk-neutral drift. Note that W (s) is now a Brownian motion

under the risk neutral measure, and is thus different to the W (s) in (2.2.1).

Our objective is to provide approximations for the current value v(xt, t) of the

derivative product and the corresponding optimal exercise and hedging strategies τ

and H:

τ : Rd × [t, T ] → 0, 1 (2.2.4)

H : Rd × [t, T ] → Rd. (2.2.5)

In the following, we appeal to the complementarity formulation of the Ameri-

can option price which is presented for example in Jaillet et al. [42]. Let L be the

related diffusion operator

L = 12 trσσ′

∂2

∂x2+ µRN

∂

∂x− r (2.2.6)

where r is the risk-free rate. Then the option value is found by solving the comple-

mentarity problem

∂v∂t + Lv ≤ 0

v − ψ ≥ 0(

∂v∂t + Lv

)

(v − ψ) = 0

(2.2.7)

for (x, s) ∈ Rd × [t, T ] with the terminal condition v(·, T ) ≡ ψ(·, T ).

2.3. Methodology 15

2.3 Methodology

To solve the complementarity problem (2.2.7) we first form a semidiscrete comple-

mentarity problem by discretising the state space but leaving time continuous. This

involves sampling the state space using QMC trials and finding a suitable approx-

imation of L. We then use standard time stepping techniques to form a system of

fully discrete LCPs. There are many methods for solving LCPs; examples include

projected successive overrelaxation (PSOR) and linear programming.

We first present and motivate each step of the algorithm separately, and then

summarise by providing a concise statement of the algorithm.

2.3.1 State space discretisation

We first consider a semidiscrete approximation to the complementarity problem

(2.2.7) in which the state space is discretised and time left continuous. This is

often called the method of lines. In the pricing problem this amounts to approxi-

mating (2.2.7) by a system of ordinary differential equations with complementarity

conditions.

The choice of a constant grid in the state space has the advantage that Crank-

Nicolson and implicit solutions can be easily considered. This seems advantageous

since, in the case of solving PDEs without complementarity conditions, the Crank-

Nicolson method is known to have a convergence rate of δt2 rather than δt for

other first order schemes. Furthermore when solving discretised complementar-

ity problems, the implicit scheme is the only time stepping method known to be

unconditionally stable (see Chapter 6 and Glowinski et al. [35]).

The choice of grid begs importance sampling considerations. That is, in order

to obtain a more accurate approximation, more grid points should be placed at

states which are more likely to be visited by the process, and at locations where the

value function has greater magnitude.

We denote the grid by X = x1, . . . , xn ⊆ Rd, and the corresponding op-

erator approximation by A. The construction of A will be considered in Section

2.3.2.

Assuming that X and A are given, we form the corresponding semidiscrete


complementarity problem

dvdt (s) +Av(s) ≤ 0

v(s) − ψ ≥ 0(

dvdt (s) +Av(s)

)′(v(s) − ψ) = 0

(2.3.1)

for s ∈ [t, T ] with terminal condition vi(T ) = ψ(xi) for each i = 1, . . . , n. Note

that v(s) is now a time-dependent vector in Rn where n is the number of grid

points.

It is also instructive to view the semi-discrete setting as a Markov chain approx-

imation to the optimal stopping problem. That is, the processX(s) is approximated

by a process restricted to the grid X . The operator A gives transition intensities on

this grid.

2.3.2 Approximating the partial differential operator

We now propose a method for specifying A in (2.3.1) for a given grid X . The

method is inspired by numerical integration, and in the European case the resulting

method will reduce to numerical integration with importance sampling. This prop-

erty is emphasised in Glasserman [33] as a favourable property of the stochastic

mesh method presented by Broadie and Glasserman [19].

We assume that the grid X has been generated using random or QMC draws

with respect to a certain density g(x). We also assume that the joint density fx,T−t

of the stochastic process is available for arbitrary initial points x and time horizons

T − t, although in principle one could adapt the following construction to the

case where the density is not known explicitly, but for example the process can be

simulated.

Denote by P (T−t) the matrix with entries

p(T−t)ij =

1∑n

k=1 fxi,T−t(xk)· fxi,T−t(xj) (2.3.2)

where the weights are given by

fxi,T−t(x) =fxi,T−t(x)

g(x). (2.3.3)

The matrix P (T−t) is a stochastic matrix, that is, a matrix with nonnegative

entries and unit row sums. We think of the entries as giving transition probabilities

between points in the grid over the horizon T − t.

2.3. Methodology 17

In the semidiscrete Markov chain setting, whereA represents transition intensi-

ties, we note that the evolution of state probabilities is given by p(s) = eA′(s−t)p(t)

where p(t) is the initial probability distribution at time t, for example it may be a

delta function in the case where the initial state is known. The matrix P (T−t) thus

gives us access to an approximation A to L on X as follows:

A ,1

T − tlogP (T−t). (2.3.4)

The matrix logarithm of P (T−t) certainly exists and is unique if the matrix

is diagonalisable and has positive eigenvalues. We find these two properties hold

in our experiments; note however that P (T−t) is in general not symmetric. We

shall see in Section 2.3.3 that instead of computing the matrix logarithm, one may

alternatively compute the matrix root corresponding to the required time step.

2.3.3 Time discretisation

Let us now discretise (2.3.1) with respect to time. We denote the approximation at

state xi and time step tk by vi,k.

We use the θ-method, standard in the numerical solution to PDEs, to discre-

tise (2.3.1). For PDE solutions, θ = 0 corresponds to the explicit method, θ = 1

corresponds to the implicit method and θ = 12 corresponds to the Crank-Nicolson

method. The latter has δt2 convergence for European problems, whereas the ex-

plicit and implicit methods exhibit δt convergence.

To implement the θ-method, we consider the vector v(k) of values at our grid

points each at time tk and discretise the first line of (2.3.1) as

v(k+1) − v(k)

δtk+A

(

(1 − θ)v(k+1) + θv(k))

≤ 0 (2.3.5)

where δtk , tk+1 − tk. Thus (2.3.1) becomes

(I + (1 − θ)Aδtk) v(k+1) − (I − θAδtk) v

(k) ≤ 0

v(k) − ψ ≥ 0(

(I + (1 − θ)Aδtk) v(k+1) − (I − θAδtk) v

(k))′ (

v(k) − ψ)

= 0.(2.3.6)

Now note that I +Aδtk = exp(Aδtk) + o(δtk). We thus define the matrices

ML = exp −θAδtk (2.3.7)

MR = exp (1 − θ)Aδtk . (2.3.8)


The approximating complementarity problem to solve is then

MRv(k+1) −MLv

(k) ≤ 0

v(k) − ψ ≥ 0(

MLv(k) −MRv

(k+1))′ (

v(k) − ψ)

= 0

(2.3.9)

for k = K − 1, . . . , 0 where the inequalities are componentwise and v(K) = ψ.

Numerically we must solve an LCP at each time step, for which the PSOR

method of Cryer [25] has been used with much success in the past. Since the

solution does not change greatly between time steps, a good starting guess for

PSOR is the solution at the previous time step. Various other methods may be

used for solving (2.3.9); for example, see Dempster and Hutton [26] for American

option pricing using linear programming in the one-dimensional case.

An error analysis of the discretisation in (2.3.5) may be undertaken along the

lines of Glowinski et al. [35] on variational inequalities or that of Kushner and

Dupuis [46] on stochastic control. Chapter 6 formulates sufficient conditions for

convergence using the framework of [35].

It turns out that the matrix logarithm does not have to be calculated explicitly

in our method; instead we may calculate roots of the matrix P (T−t) corresponding

to the time step and implicitness parameters. In particular we have

ML =(

P (T−t))−θδtk/(T−t)

(2.3.10)

MR =(

P (T−t))(1−θ)δtk/(T−t)

. (2.3.11)

We prefer to use the matrix root because we have found it to be a quicker and

more robust operation in Matlab than the matrix logarithm. If one would choose

to compute the logarithm however, one would have access to a varying time step

without performing any extra computations.

There are many methods available for evaluating matrix functions, as detailed

in Golub and Van Loan [36]. The general method suggested involves Schur decom-

position in combination with Parlett’s algorithm, which computes general functions

of an upper triangular matrix. Matrix functions can also be computed using eigen-

decomposition, which is the method used by Matlab to compute general matrix

powers. We note that the structure of the matrix P (T−t) may mean that more effi-

cient methods are available for computing matrix roots and logarithms; it is not the

purpose of the current research to investigate such methods however.

2.3. Methodology 19

We now highlight the importance of using the matrix logarithm or root, as op-

posed to constructing P (δt) directly (the latter being more attractive computation-

ally). The intuition for this importance is that P (δt) does not produce consistent

transition probabilities over longer time horizons as in (2.3.12). We demonstrate

the difference between the two constructions in Figure 2.3.1 for a one-dimensional

example and a random grid. In particular, when δt is too small compared to the

separation of grid points, the solutions become distorted. This problem is more

pronounced in higher dimensions due to the larger average separation between grid

points.

−4 −3 −2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

log(S0)

v 0

−4 −3 −2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

log(S0)

v 0

Figure 2.3.1: Random grid valuation of an American put option on a single asset

with expiry 1, strike 1 and 100 asset points, using transition matrices P (0.01) and(

P (1))0.01

, respectively (dots). The plots are in the log domain. Also plotted are

values computed to high accuracy (solid lines) using a standard finite difference

method.

Remark 2.3.1 It is clearly more efficient if the matrices ML and MR need be

calculated only once; hence the choice of a constant time step δtk ≡ δt seems

convenient. We also note that, given a small enough δt, ML and MR should be

approximately sparse in that most elements can be set to zero without affecting

the solution significantly. Using this observation can dramatically improve the

efficiency of the solution procedure.

Remark 2.3.2 As already noted above, an important property of this construction

in the European setting is that the time stepping reconstructs numerical integration


with importance sampling function g. The reconstruction is realised as follows:

vi(t) =

(

K∏

k=1

e−rδtM−1L MR

)

v(K)

= e−r(T−t) exp(A)v(K)

= e−r(T−t)n∑

j=1

ψ(xj)p(T−t)ij (2.3.12)

where vi(t) is the value in state xi at initial time t and p(T−t)ij is defined in (2.3.2).

The last line of the equation is precisely QMC integration of the payoff ψ with

importance sampling function equal to the grid density g. Note that in caseML and

MR are constructed from the matrix logarithm, (2.3.12) holds only asymptotically

as δt→ 0.

Equation (2.3.12) also shows that the calculation of the European price on the

grid may be carried out without time stepping, given that the transition probabilities

p(T−t)ij are available. Thus, using the European price as control variate is a faster

operation than would normally be expected.

2.3.4 Randomisation

The QMC grids we have proposed are deterministic; however perturbing these

points randomly allows us to observe the behaviour of solutions for a random se-

lection of QMC grids, and thus to obtain estimates of the bias and standard error

of solutions. The use of such methods is surveyed in Owen [59] for integration

problems. The importance of randomised QMC is also emphasised in Glasserman

[33].

When using Sobol’ points and a normal density for example, one first gener-

ates the Sobol’ points, then applies the inverse normal distribution function to the

points. In order to realise randomised QMC points, one perturbs the Sobol’ points

modulo one by a random factor before applying the inverse normal distribution

function.

Suppose S = (si) is our sequence of n Sobol’ points, and Uj is a sequence of

random variables uniformly distributed on the unit cube [0, 1]d. We then realise the

jth randomised Sobol’ sequence as

Sj = (sj,i)i∈N = (si + Uj mod 1)i∈N. (2.3.13)

2.4. Experimental setup and details 21

We refer to grids obtained in this way as randomised QMC (RQMC) grids.

2.3.5 Summary of procedure

We now present a concise statement of the proposed procedure as Algorithm 2.3.1.

We let vi,j denote the solution at initial time t and state xi in the jth RQMC exper-

iment. For the statement of the algorithm we assume a fixed number of grid points

n and a constant time step δt = (T − t)/K where K is the number of time steps.

Algorithm 2.3.1 The proposed irregular grid algorithmfor j = 1, . . . , J do

Generate a RQMC grid X

Compute the transition matrix for expiry P (T−t)

Compute the matrix root(

P (T−t))1/2K

(Crank-Nicolson)

Solve the LCPs (2.3.9)

Let vi,j be the solution at initial time t for state xi

end forfor initial states of interest xi do

Estimate the solution as vi = 1J

∑

j vi,j

Estimate the standard error as εi =(

1J−1

∑

j(vi,j − vi)2)1/2

.

end for

2.4 Experimental setup and details

We now use the algorithm presented in Section 2.3.5 to estimate prices of multi-

asset options. We first present a detailed exposition of the setting, experimental

procedure and various considerations. Numerical results are presented in the next

section.

2.4.1 Specification of dynamics

Suppose our American option is based on d assets following a correlated geometric

Brownian motion where the risk-neutral dynamics in the log domain are given by

dX =(

r11 − δ − 12 diag(Σ)

)

dt+R′dW (2.4.1)


and r is the risk-free rate, 11 is the d-vector of ones, δ = (δ1, . . . , δd) is the vector

of dividend rates, Σ = (ρijσiσj) is the covariance matrix of the Brownian motions

and R′R is its Cholesky decomposition. The operator L in this setting is just the

multidimensional Black-Scholes operator given by

L = 12

d∑

i,j=1

ρijσiσj∂2

∂xi∂xj+

d∑

i=1

(r − δi −12σ

2i )

∂

∂xi− r. (2.4.2)

2.4.2 Elimination of drift

In order to facilitate reuse of the matrix roots, we first reformulate the problem so

that the process has zero drift. We introduce the change of variables

X0(s) = X(s) − (s− t)µ, (2.4.3)

where µ is the risk-neutral drift; for example in (2.4.1) we have µ = r11 − δ −12 diag(Σ). The new process X0 has zero drift and the covariance Σ is unchanged:

dX0(s) = RdW (s). (2.4.4)

The payoff under the reformulation is

ψ0(xi, s) = ψ (xi + (s− t)µ) . (2.4.5)

One may also eliminate a deterministic, time-dependent risk-neutral drift by sub-

tracting∫ st µ(u)du in (2.4.3).

2.4.3 Grid specification

We consider normal RQMC grids as suggested in Section 2.3.4; thus the grid den-

sity is multivariate normal. We now discuss parameter selection for the grid den-

sity.

Importance sampling considerations tell us that the most efficient sampling is

given by the density of the process itself; thus using a constant grid we cannot

provide the most efficient importance sampling at all times. However, given the

restriction to a constant grid, we can still provide an acceptable importance sam-

pling.


As outlined in Evans and Swartz [28], the rate of convergence for importance

sampling of normal densities using normal importance sampling functions is most

damaged when the variance of the importance sampling function is less than that

of the true density. Conversely, convergence rates are not greatly affected when

the variance of the importance sampling function is greater than that of the true

density.

The situation we should try to avoid is that the process has a significant prob-

ability of lying in the “tails” of the grid density. A further consideration is the

minimisation of boundary effects on the solution. This suggests that the grid co-

variance should be larger than the covariance of the process.

These considerations lead us to set the grid mean to the initial state xt and the

grid covariance to be a multiple α of the grid density at expiry for some trial values

α = 1.0, 1.5, 2.0. Owing to the reformulation (2.4.3), this ensures that the grids

are centered at the process mean for all times. We further ensure that the initial

state is included in the grid.

Summarising, we suggest the parameters

µg = xt (2.4.6)

Σg = αΣ(T − t). (2.4.7)

The first grid point in the jth RQMC experiment is x1 = µg and the (i+ 1)th grid

point is generated as

xi+1 = µg +R′g

(

Ψ−1(sj,i,1) · · ·Ψ−1(sj,i,d)

)′(2.4.8)

where Ψ−1 is the standard normal inverse function, R′gRg is the Cholesky decom-

position of Σg and sj,i,k is the kth component of sj,i.

An example of a normal Sobol’ grid in two dimensions is shown in Figure

2.4.1. It should be noted however that the advantage of using an irregular grid is

realised in dimensions of at least three.

2.4.4 Reuse of roots for similar processes

Given that generating matrix roots is an expensive operation compared to the final

time stepping procedure, it is of interest to know under which conditions these ma-

trix roots can be reused for related problems; for example, problems with different

parameters.


−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Figure 2.4.1: Example of normal QMC grid in 2 dimensions with 500 points.

Clearly a single matrix root can be reused for as many different payoff func-

tions as required, but we also show how it can be reused for processes with different

risk-neutral drifts and covariances. To answer this question for diffusion processes

with zero drift we provide the following result.

Lemma 2.4.1 Suppose that P (T−t) is the transition matrix corresponding, through

(2.3.2) and (2.3.3), to the grid X = x1, . . . , xn, respective importance sampling

weights g1, . . . , gn, horizon T − t and an d-dimensional Brownian motion with

covariance Id. Suppose further that P (T−t) is the transition matrix corresponding

to the grid

Y = y1, . . . , yn = R′x1, . . . , R′xn, (2.4.9)

importance sampling weights g1, . . . , gn, horizon T − t and an d-dimensional

Brownian motion with positive definite covariance Σ = R′R.

Then

P (T−t) = P (T−t). (2.4.10)

Proof. Let fx,T−t and hx,T−t be the densities at expiry from starting point x,

expiry T − t and corresponding to covariance I and Σ, respectively. The densities

from xi to xj in grid X and from yi = R′xi to yj = R′xj in grid Y are respectively

fxi,T−t(xj) = |2π(T − t)Id|−1/2 exp

− 12(T−t) (xj − xi)

′(xj − xi)

hyi,T−t(yj) = |2π(T − t)Σ|−1/2 exp

− 12(T−t) (xj − xi)

′RΣ−1R′(xj − xi)

= |2π(T − t)Σ|−1/2 exp

− 12(T−t) (xj − xi)

′(xj − xi)

,


which are equal up to the constant factor |Σ|1/2. Given the latter observation and

that the weights are equal in both cases, we conclude that the normalised entries

p(T−t)ij and p(T−t)

ij obtained through (2.3.2) and (2.3.3) are also equal.

Remark 2.4.1 One may ask whether the weights gi specified in Lemma 2.4.1 are

indeed appropriate for the grid Y . That is, whether (2.3.3) leads to a standard

importance sampling procedure for Y with these weights. We answer this question

by comparing the grid densities.

Suppose that the density gX was used to generate the grid X using random

sampling. So that for every S ⊂ Rd,

P (x ∈ S) =

∫

SgX(x)dx.

Applying the transformation x 7→ y = R′x leads us to conclude that the grid Y

consists of points generated randomly from some density gY satisfying, for each

S ⊂ Rd,

P (y ∈ R′S) =

∫

R′SgY (y)dy

=

∫

SgY (R′x)

∣

∣|R|∣

∣dx (2.4.11)

by the multivariate substitution formula where R′S = R′x : x ∈ S and∣

∣|R|∣

∣ =

abs(detR). But since y = R′x, P (y ∈ R′S) = P (x ∈ S), and since (2.4.11)

holds for all S ⊂ Rd, we conclude that

gX(x) =∣

∣|R|∣

∣gY (R′x). (2.4.12)

Finally, note that the averaging taking place in (2.3.3) implies that the weights

gX(x), being proportional to gY (R′x), are appropriate for importance sampling

with respect to the grid Y .

Remark 2.4.2 A time-dependent scaling of the covariance can also be incorpo-

rated by using the matrix logarithm, and constructing the time stepping matrices

through (2.3.7) and (2.3.8) rather than (2.3.10) and (2.3.11).


2.4.5 Low-biased estimate

As is common practise in the American option pricing literature, a low biased

estimate may be obtained by taking an exercise rule implied by the pricing method

and determining the expected value of using this rule using out-of-sample paths.

A natural approximation to the optimal exercise rule using the pricing results

of the irregular grid method is to take the implied rule of the nearest neighbour in

the grid at the closest time. Specifically one may define the exercise rule for grid

points to be

τ(xi, tk) ,

1 if v(k)i = ψi

0 otherwise(2.4.13)

and for general points x ∈ Rd and times s ∈ [t, T ]

τ(x, s) ,

1 if v(k)i = ψi

0 otherwise(2.4.14)

where k = argminj |s− tj| and i = argminj ||x− xj|| : xj ∈ Xk.

This rule is easily implemented and can also be adapted to the case where we

have several different grids. In this case one could base the exercise rule on a vote

between grids. One could also implement weighted schemes with respect to x and

t rather than using nearest neighbour rules.

2.4.6 High-biased estimate

Whereas applying an exercise rule to out-of-sample paths leads to a low-biased

estimate of the option value, simulating the cost of a hedging strategy leads to

a high-biased estimate. The latter may be seen as follows: the optimal hedging

strategy enables the seller of the option, given a cash amount equal to the value

of the option at the initial time t, to perfectly reproduce the payoff. A suboptimal

strategy however will on average require a larger initial cash amount, thus the cost

of a suboptimal hedging strategy is on average higher than the true option value.

A formal demonstration can be given in terms of the dual formulation for

American option pricing (see Rogers [64], Haugh and Kogan [37]) in which one

minimises the cost of hedging by minimising an objective function over martin-

gales. Since the value of our hedging strategy is a martingale, it corresponds in

general to a suboptimal martingale, and thus a high-biased estimate.


In practise, obtaining an upper bound in the way we suggest requires knowl-

edge of the optimal exercise rule. Since we only have an estimate of this, the cost

of the hedging strategy may not be purely upward biased. We find however that

one can approximate the optimal exercise strategy far more accurately than one

can approximate the optimal hedging strategy. We shall see in Section 2.5 that

experimental results support this statement.

In the literature on American options there is little said about the practicalities

of hedging in a high dimensional setting. The difficulty with using an approach

such as LSM is that the method does not naturally form approximations to the

value function from which derivatives can be estimated. One can form a hedging

strategy by evaluating prices at states perturbed in each underlying; this demands

the calculation of many additional option prices at each time step, each calcula-

tion being expensive in a high-dimensional setting. Furthermore one must be very

careful with partial derivative estimates obtained from differencing stochastic point

estimates; in particular the point estimates must be sufficiently accurate and the

perturbations must be well-chosen with respect to the (unknown) curvature of the

value function.

A solution provided by the irregular grid method involves estimates of the price

not only at the current state, but at all states in the grid. This allows one to extract

derivative estimates using value information from nearby points in the grid; for

example using partial derivatives implied by a local linear regression. Indeed the

irregular grid method provides derivative information as a by-product.

2.4.7 Benchmarks

There are few benchmark results for high-dimensional American options. Broadie

and Glasserman [19] provide 90% confidence intervals for American call options

on the maximum of five assets with nine exercise opportunities and the geometric

average of five and seven assets with ten exercise opportunities using their stochas-

tic mesh method. Longstaff and Schwartz [50] price the Broadie and Glasserman

maximum options using the LSM method.

Stentoft [68] uses the binomial method of Boyle et al. [13] and the LSM me-

thod to price put options on the arithmetic average, geometric average, maximum

and minimum of three and five assets. Broadie and Glasserman [18] and Fu et

al. [31] provide benchmark results for options over five assets with three exercise


opportunities. Finally, Rogers [64] and Haugh and Kogan [37] use the dual formu-

lation to price a number of different American options.

A useful result involving options on the geometric average of several assets

is that this problem can be easily reduced to an option pricing problem in one

dimension. Suppose that the risk-neutral dynamics in the log domain are given by

(2.4.1), and the payoff function

ψ(s) =

(

K −(

∏

si

)1/d)+

(2.4.15)

where x+ denotes the positive part of x, K is the strike price and d is the number

of assets. Then using Ito’s lemma one finds that the price is the same as that of a

vanilla put on the asset with log price Y where Y (t) = 1d

∑di=1Xi(t) and

dY (s) =1

d

d∑

i=1

dXi(s) (2.4.16)

= µds+ σdW (s). (2.4.17)

The parameters of the diffusion are given by

µ = r −1

2d

d∑

i=1

σ2i (2.4.18)

σ2 =1

d2

d∑

i=1

d∑

j=1

Rij

2

. (2.4.19)

Using this we find that an accurate price for the geometric average European

option in the Stentoft setting is 1.159, and the Bermudan and American prices are

1.342 and 1.355, respectively. Note that the difference in early exercise premium

between the Bermudan, which allows ten exercise opportunities, and American

prices is about 6%.

2.5 Experimental results

Our experiments are conducted in a Matlab environment and are based on the five-

dimensional examples of Stentoft [68]. Specifically we consider five stock pro-

cesses driven by correlated Brownian motions for put options with four different

2.5. Experimental results 29

payoff functions. The method we use for valuation is that of Section 2.3. Our

programs are mostly script-based but some computationally intensive routines, for

example the PSOR code, have been implemented in C.

We are given initial stock prices Si(0) = 40 for each i, the correlations between

log stock prices are ρij = 0.25, i 6= j, and volatilities are σi = 0.2 for all i, the

risk-free interest rate is fixed at r = 0.06, the expiry is T = 1 and we use K = 10

time-steps.

We generate 50 RQMC normal grids as detailed in Section 2.3.4 using the

parameter values α = 1, 1.5 and 2, respectively (these were found to give the

best rates of convergence). The number of grids need not be so high in practise,

depending on the accuracy required. The vector of initial stock prices x0 was

always included in the grid.

The payoff functions considered correspond to put options on the arithmetic

mean, geometric mean, maximum and minimum, respectively,

ψ1(s) =(

K − 1d

∑

si

)+ψ2(s) =

(

K − (∏

si)1/d)+

ψ3(s) =(

K − max(si))+

ψ4(s) =(

K − min(si))+.

(2.5.1)

Figures 2.5.1–2.5.3 show the convergence behaviour of the irregular grid me-

thod where the implicitness parameter is θ = 0, 12 and 1, respectively, and for grid

sizes up to 1000. For the constrained solutions, we see that convergence is usually

fastest for α = 1.5, the algorithm reaching a fairly stable value for n = 1000 for

all but the maximum option.

The solutions for the arithmetic and geometric average options appear to con-

verge to Stentoft’s solutions for Bermudan options with ten exercise opportunities

in the explicit case. For the Crank-Nicolson and implicit cases, the solutions appear

to converge to a higher value.

The previous observation may be explained as follows. In the explicit case, our

method calculates the price of a Bermudan option with ten exercise opportunities,

just as in the case of Stentoft (provided we use ten equal time steps). This is because

the explicit formulation takes the maximum of the intrinsic and continuation values

at each exercise opportunity, and because we use exactly ten time steps. One can

still see this Bermudan price as an approximation to the true American price, which

we calculated previously to have an early exercise premium approximately 6%


higher than the Bermudan price in the case of the geometric average put option

over five assets.

In the Crank-Nicolson and implicit cases however, we cannot interpret the so-

lution as approximating a Bermudan option due to the implicitness of the formula-

tion. We can only say that as δt→ 0, the solution should converge to the American

price. In the Crank-Nicolson case we suspect that the convergence is faster than

in the implicit case (drawing a parallel with the unconstrained problem), and so

we can think of our Crank-Nicolson solution as being close to our best possible

approximation to the true American option value, given that we use ten equal time

steps. We thus stress that the convergence of the Bermudan price does not require

δt→ 0, but the convergence of the American price does.

The fastest convergence rate in Figures 2.5.1–2.5.3 is achieved with α (the ratio

of grid density to process density) being 1.5. We thus present in Tables 2.5.1 and

2.5.2 some results and comparisons for Bermudan and American option prices,

respectively, using the irregular grid method with a normal grid and α = 1.5.

Given the previous discussion, we take our explicit solutions to be approximations

to the Bermudan problem, and the Crank-Nicolson solutions to be approximations

for the American problem.

Tables 2.5.1 and 2.5.2 also show out-of-sample results for LSM and the irregu-

lar grid methods. These are estimates of the expected value, under the risk-neutral

measure, of using the implied exercise strategy. We implement the LSM method

ourselves, as specified in Stentoft [68], to obtain out-of-sample values for the LSM

algorithm (these results are not given in [68]). Our LSM implementation also re-

produced (up to a statistically insignificant difference) the in-sample LSM results

given in [68]. For details of how out-of-sample paths are used in the LSM method

to obtain low-biased estimators, we refer the reader to Longstaff and Schwartz [50].

We remark that the values obtained from the irregular grid method are higher

than those produced by the LSM algorithm, although this is not statistically signif-

icant except in the case of the minimum option. The OS results are also higher for

all but the maximum option.

For the more problematic cases of the maximum and minimum options, we see

that convergence is slower. In the case of the maximum it is not clear with 1000

grid points what an appropriate estimate should be. It is also not clear whether the

convergence in our case for the explicit method agrees with the value obtained by


100 200 300 400 500 600 700 800 900 10000.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

100 200 300 400 500 600 700 800 900 10001.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

Arithmetic average Geometric average

100 200 300 400 500 600 700 800 900 10000.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

100 200 300 400 500 600 700 800 900 1000

5.4

5.5

5.6

5.7

5.8

5.9

6

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

Maximum Minimum

Figure 2.5.1: Average QMC grid valuation over 50 normal grids with α = 1.0 (cir-

cles), α = 1.5 (squares), α = 2.0 (diamonds) of European (solid lines) and Amer-

ican (dotted lines) put options over five assets using the explicit method (θ = 0.0)

and ten time steps. Stentoft’s Bermudan LSM solutions are drawn as horizontal

lines.


100 200 300 400 500 600 700 800 900 10000.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

100 200 300 400 500 600 700 800 900 10001.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)


100 200 300 400 500 600 700 800 900 10000.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

100 200 300 400 500 600 700 800 900 10005.4

5.5

5.6

5.7

5.8

5.9

6

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

Maximum Minimum

Figure 2.5.2: Average QMC grid valuation over 50 normal grids with α = 1.0

(circles), α = 1.5 (squares), α = 2.0 (diamonds) of European (solid lines) and

American (dotted lines) put options over five assets using the Crank-Nicolson me-

thod (θ = 0.5) and ten time steps. Stentoft’s Bermudan LSM solutions are drawn

as horizontal lines.


100 200 300 400 500 600 700 800 900 10000.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

100 200 300 400 500 600 700 800 900 10001.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)


100 200 300 400 500 600 700 800 900 10000.1

0.15

0.2

0.25

0.3

0.35

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

100 200 300 400 500 600 700 800 900 10005.4

5.5

5.6

5.7

5.8

5.9

6

Grid points (n)

Est

imat

ed o

ptio

n va

lue

(v0)

Maximum Minimum

Figure 2.5.3: Average QMC grid valuation over 50 normal grids with α = 1.0 (cir-

cles), α = 1.5 (squares), α = 2.0 (diamonds) of European (solid lines) and Amer-

ican (dotted lines) put options over five assets using the implicit method (θ = 1.0)

and ten time steps. Stentoft’s Bermudan LSM solutions are drawn as horizontal

lines.


Option Exact Binomial LSM LSM OS Normal Normal

type (low- grid grid OS

biased) (low-biased)

Arith. - 1.235 1.241 1.231 1.246 1.238

Average (0.0006) (0.0006) (0.004) (0.005)

Geom. 1.342 1.340 1.348 1.335 1.350 1.345

Average (0.0006) (0.0007) (0.004) (0.005)

Maximum - 0.230 0.275 0.268 0.276 0.233

(0.0004) (0.0004) (0.008) (0.002)

Minimum - 5.841 5.815 5.816 5.847 5.821

(0.0012) (0.0014) (0.009) (0.013)

Table 2.5.1: Comparison of Bermudan price estimates (θ = 0) with ten exer-

cise opportunities. The grid estimates are the average price taken over 50 normal

RQMC grids with size 1000, with α = 1.5 and using ten time steps. The bino-

mial method of Boyle et al. [13] was used with Richardson extrapolation. The OS

(out-of-sample) columns give the value of the exercise strategy implied by the 50

grid solutions, calculated by taking the mean value over 100,000 simulated paths.

The binomial and LSM prices are given in [68] and the OS prices for LSM are

computed by running the LSM method 20 times each with 100,000 out-of-sample

paths. The exact price given in the first column is the numerical solution to the

equivalent one-dimensional problem. Standard errors are shown in brackets.


Option Exact Normal grid Normal grid Hedged

type American American American OS American OS

(low-biased) (high-biased)

Arith. - 1.257 1.243 1.363

Average (0.004) (0.004) (0.004)

Geom. 1.355 1.360 1.348 1.462

Average (0.004) (0.005) (0.004)

Maximum - 0.295 0.267 0.504

(0.009) (0.002) (0.006)

Minimum - 5.862 5.789 6.355

(0.009) (0.012) (0.010)

Table 2.5.2: Comparison of American price estimates (θ = 0.5). The grid esti-

mates in the third and fourth columns are the average price taken over 50 normal

RQMC grids with size 1000, with α = 1.5 and using ten time steps. The OS

(out-of-sample) column gives the estimated value of the implied exercise strategy,

calculated by taking the mean value over 100,000 simulated paths and using 50

time steps. The hedged column gives the average cost of the hedging strategy ob-

tained as a by-product of a single price computation; it is implemented using the

results of a single grid solution, 50 time steps and computes the hedge as detailed

in Section 2.4.6. In particular note that we have used a different time step in the

OS exercise and hedging simulations than in computing the grid solutions. The

exact price given in the first column is the numerical solution to the equivalent

one-dimensional problem. Standard errors are shown in brackets.


Stentoft. These are cases where the grid could be adapted to the payoff function as

well as to the process itself; such extensions are left for future investigation.

In Table 2.5.1 it is encouraging to see that the irregular grid prediction for the

geometric average option is very accurate as compared to the benchmark. The

exercise strategy performs well for the arithmetic and geometric average options,

but not for the more problematic maximum and minimum payoffs.

As detailed in Section 2.4.6, our method yields a hedging strategy as a by-

product; thus simulation of a hedging strategy can be done quickly and efficiently.

Using the implied hedging strategy of a single grid, and taking 20 nearest neigh-

bours for the delta estimation, we obtain the results shown in the last column of

Table 2.5.2. It is clear that the hedging errors are much larger than the exercise

errors; this may be expected given that the exercise rule is a function having only

two possible values, whereas the hedging rule takes values in Rd. The hedging

strategy used is naive in that the results of only a single grid solution are used. It

could probably be improved by using information from different grid solutions.

The most time-consuming operation in the irregular grid method is the compu-

tation of the matrix root. Some timings for computing matrix roots in Matlab 6.1

on a PIII 866MHz machine are presented in Table 2.5.3. It should be noted that

the time does not depend strongly on the order of the root, so that square root and

tenth root operations for example take about the same amount of time. The time

taken for the construction of the matrix P (T−t) is seen to be small compared to the

root operation.

Although the matrix root operation is time-consuming for large values of n,

it should be noted that once a root has been computed for a single normally dis-

tributed grid, it can subsequently be used for valuing options on a large class of

diffusion processes with arbitrary payoff functions without the need for recompu-

tation.

2.6 Conclusions

We have proposed a new method for finding the value of American and Bermu-

dan options in a high-dimensional setting. Central to this method is the use of an

irregular grid over the state space and an approximation of the partial differential

operator on this grid.

2.6. Conclusions 37

Size

P

(n)

Memory

full

(MB)

Memory

sparse

(MB)

Time

for P

(sec)

Time

for P 1/10

(sec)

Prop

P 1/10

nonzero

Time-

stepping

(sec)

500 2.0 0.6 1 22 0.190 0.5

1000 8.0 1.8 5 200 0.147 1.3

1500 16.0 3.3 12 750 0.123 2.0

2000 32.0 5.1 22 2000 0.106 2.9

2500 50.0 7.1 37 4000 0.094 3.8

3000 72.0 9.1 55 7200 0.084 4.9

Table 2.5.3: Timings and storage requirements for the irregular grid method us-

ing Matlab 6.1 with a PIII 866MHz processor with 512 MB RAM, matrix entries

stored in double precision (8 bytes per entry). The sparse matrices are formed by

eliminating all entries smaller than 5× 10−4 and renormalising. The time stepping

column gives the total time to complete 10 time steps, using the sparse matrix and

the explicit method. Note that sparse matrices were not used for any experiments

in this chapter, the information provided rather serves to illustrate the complexity

of the method as n increases. The second-to-last column gives the proportion of

nonzero entries in the sparse matrices, an important consideration for computa-

tional complexity. Note that MB denotes 106 bytes in this context.


In our analysis we allow any grid which is generated using MC or QMC trials

with respect to a known density function. Once the Markov chain approximation

has been obtained, we use the transition probability matrix to form a semidiscrete

approximation to the partial differential operator corresponding to this Markov

chain. This is done through taking a logarithm of the transition probability ma-

trix; however solving the fully discrete problem only requires computing a certain

root of the matrix related to the time step and implicitness parameters, at the cost

of an extra approximation error.

An important aspect of the proposed method is the absence of any requirement

to specify basis functions for approximating the value function or exercise strategy.

Indeed the only specification needed is a grid density, although asymptotically even

this choice is not critical. Furthermore, convergence in the Bermudan case should

require asymptotics in only one parameter, namely the number of grid points. In the

American case one also requires δt → 0. These aspects set the root method aside

from the LSM method where the specification of basis functions plays a critical

role in the success of the method, and convergence involves asymptotics in two

parameters in the Bermudan case, namely the number of basis functions and the

number of paths, these two parameters producing opposite biases.

The irregular grid solution gives price estimates at all points in the grid. This

is useful if one requires partial derivative information, for example when hedging.

Partial derivatives can be easily estimated from the solution by preforming a linear

regression using values from neighbouring points.

Our experiments suggest that the irregular grid method has very good conver-

gence properties, especially when the grid density is related to the density of the

process itself. In particular, the grid density should have a larger variance than

the process; for a geometric Brownian motion process in five dimensions it was

found that a ratio of 1.5 gave a good rate of convergence, although (slower) con-

vergence was also observed for ratios of 1.0 and 2.0. Convergence of estimates for

the maximum option was not clear with grids of up to 1000 points.

The numerical results obtained largely agree with those of Stentoft [68]. We

find that the early exercise premium is increased by about 6% for the examples

he considers when allowing a continuum of exercise opportunities rather than only

ten. We also find that the exercise strategies implied by the LSM method produce

significantly lower values (statistically) than the LSM price implies, except in the

2.6. Conclusions 39

case of the minimum option; this is an indication that out-of-sample paths should

be used in simulation methods — in this way the price obtained corresponds di-

rectly to the average value of the implied exercise strategy. This suggests that

one should be careful in higher dimensions when applying the recommendation of

Longstaff and Schwartz [50] to save time by only using in-sample paths.

A possible variance reduction technique is to adjust the transition probabilities

according to the empirical density of the grid points rather than the density used for

generation of the grid. Adjustment may be done after constructing the transition

matrix for example using quadratic programming to improve local consistency in

the sense of Kushner and Dupuis [46], but may also take inspiration from the liter-

ature on nonparametric analysis. These and other possible refinements are reserved

for future investigation.

Results relating to the convergence of this algorithm are provided in Chapter

6. Sufficient conditions are provided in that chapter, although it is still to be de-

termined whether these conditions are satisfied for the method presented in this

chapter.

Chapter 3

A Method Using Local QuadraticApproximations

3.1 Introduction

Recent literature tends to support the widely-held belief that pricing American op-

tions in a high-dimensional setting is a highly nontrivial task. Methods proposed

by Carriere [20], Longstaff and Schwartz [50], Rogers [64] and Haugh and Kogan

[37] give good results in many situations, but are sensitive to the choice of basis.

Chapter 2 (also published as Berridge and Schumacher [8, 7, 9]) proposes a method

which does not require the choice of a basis, as do Bally and Pages [3].

The difficulty with primal methods in a high-dimensional problem is deter-

mination of the expected continuation value. This is essentially an integral with

respect to the risk neutral process and the optimal stopping rule. The latter poses

the most difficulty since it is itself a high-dimensional function which must be ap-

proximated.

In classical finite difference methods the expected continuation value is deter-

mined through an approximation to the operator on a regular grid. In the irregular

grid setting this approximation becomes more difficult, mainly due to instabilities

in the approximation. The construction of stable approximations will be investi-

gated in this chapter through using a local polynomial representation of the value

function.

Once a stable approximation is found, one has access to finite difference tech-

41

42 Chapter 3. Local Quadratic Approximations

niques for time stepping, including the explicit, Crank-Nicolson and implicit meth-

ods. In general, these θ-methods lead to linear complementarity problems (LCPs),

which can be solved using the projected successive overrelaxation (PSOR) method

introduced by Cryer [25] or by linear programming, for example.

It is of interest to note the connection between the methods presented in this

chapter and the literature on mesh-free methods (for an overview see Liu [48]).

In particular the method of moving least squares provides an approach to partial

differential equation (PDE) solution which is similar to that found in this chapter in

that an irregular grid is used on which to calculate local quadratic approximations.

The essential difference is that our approach does not use weak forms, and thus

does not require integration of basis functions. It seems the method we present is

more sensitive to the structure of the grid, producing unstable approximations in

some circumstances. The MLS method is applied to option pricing in Chapter 5.

The remainder of this chapter is organised as follows. In Section 3.2 we in-

troduce the setting, the problem and a review of irregular grid methodology from

Chapter 2. Section 3.3 introduces the local quadratic approximation method; Sec-

tions 3.4 and 3.5 present results using this method for regular and irregular grids,

respectively. Finally Section 3.6 presents conclusions and ideas for future research.

3.2 The pricing problem

3.2.1 Formulation

We consider an American option at time t on d underlying assets with values

X(s) = (X1(s), . . . , Xd(s))′ at time s ∈ [t, T ], payoff ψ(X(s), s) and expiry

T . The assets follow the diffusion

dXi(s) = µi(X(s), s)ds + σi(X(s), s)dW (s) (3.2.1)

for i = 1, . . . , d and where X(t) is known, dW (s) are the increments of a standard

d-dimensional Brownian motion and the µi and σi are measurable with respect to

the filtration generated by the Brownian motion.

The price of an American option, giving the long party the right to receive the

payoff ψ(X(s), s) at any time s ∈ [t, T ], is given in the primal formulation as

v(X(t), t) = supτ∈T

EQ

X(t)

(

e−r(τ−t)ψ(X(τ), τ))

(3.2.2)

3.2. The pricing problem 43

where T is the set of stopping times on [t, T ] with respect to the natural filtration,

the expectation is taken with respect to the risk-neutral measure Q, and the initial

value is X(t).

This can be reformulated as the following complementarity problem

∂v∂t + Lv ≤ 0

v − ψ ≥ 0(

∂v∂t + Lv

)

(v − ψ) = 0

(3.2.3)

for (x, s) ∈ Rd × [t, T ] with the terminal condition v(·, T ) ≡ ψ(·, T ). Here L is

the Black-Scholes operator implied by the diffusion.

3.2.2 Discretisation

One way to discretise the complementarity formulation is to sample the state space.

This is the approach taken in finite difference methods, however the traditional

grid approach is not suitable for high-dimensional problems due to the curse of

dimensionality. We thus consider an irregular sampling of the state space on which

to approximate the problem.

Suppose now that we are given some sampling of the state space, X = (x1,

. . . , xn) ⊂ Rd. We do not concentrate on the properties of the sampling, but we

may assume it is a sequence of low discrepancy in the sense of Niederreiter [58]

or low distortion in the sense of Bally and Pages [3]. These quantities may be best

measured in terms of the terminal distribution of the process, which in our setting

is a multivariate normal distribution.

Having taken such a sample we approximate the complementarity problem

(3.2.3) by the new complementarity problem

dvdt +Av ≤ 0

v − ψ ≥ 0(

dvdt +Av

)′(v − ψ) = 0

(3.2.4)

where v is an n-vector and the ith component of ψ is ψ(xi, t). The matrixA should

approximate L on our grid X in that

(Av(t))i ' (Lv)(xi, t). (3.2.5)

Let us write v as v = (v(1) · · · v(n))′ where each v(i) : [t, T ] → R.


3.2.3 Nearest neighbours

Just as in traditional finite difference methods, a great deal of efficiency can be

gained by only considering local interactions. We thus use nearest neighbour sets

on which to construct local approximations to L.

The kth-nearest neighbour function Nk,X : 1, . . . , n → 1, . . . , n for some

set of points X is then defined as

Nk,X (i) , j : ‖x− xi‖ ≤ ‖xj − xi‖ for exactly k different x, x ∈X .

Note that N1,X (i) = i, that is xi is the nearest neighbour of xi in this definition.

Further let Ni = Nj,X (i)j=1,...,k be the ordered set of the k nearest neighbours

for each i. For brevity we denote the jth nearest neighbour of point xi as

x(j)i ≡ Nj,X (i).

In addition to considering other points as neighbours, we may also allow bound-

ary points to be neighbours. Thus in some situations we use the extended nearest

neighbour function Nk,X : 1, . . . , n → 1, . . . , n+ 2d where the 2d extra

points are projections of xi onto the boundaries.

3.3 Methodology

3.3.1 Approximating the differential operator

We consider now the construction of a direct approximation to A with respect to

a grid X . Let us form this approximation by assuming that v is approximately

locally quadratic about each grid point xi. This is justified since we know that v

is almost everywhere continuously differentiable, and convenient since we wish to

approximate the effect of the second order operator L.

We write L as

L = α0,0 +d∑

j=1

αj,0∂

∂xj+

d∑

j=1

j∑

k=1

αj,k∂2

∂xj∂xk

for some αj,k ∈ R.

3.3. Methodology 45

Now let us introduce the quadratic interpolant v(i) : Rd → R at grid point xi

as

v(i)(x) = a(i)0,0 +

d∑

j=1

a(i)j,0xj +

d∑

j=1

j∑

k=1

a(i)j,kxjxk (3.3.1)

where the a(i)j,k are chosen so that v(i)(x

(j)i )(s) = v(j)(t) for all s ∈ [t, T ] whenever

j ∈ Ni. By finding this interpolant for each i we have an approximation for v(x, t)

in a neighbourhood of all our grid points. Letting η = 12

(

d2 + 3d+ 2)

(the num-

ber of parameters in each v(i)), we can determine a unique quadratic interpolant

for each i. Denoting the kth component of the jthe neighbour of xi by x(j)i,k , the

coefficients of the ith interpolant are

a(i)(t)

≡(

a(i)0,0, a

(i)1,0, . . . , a

(i)d,0, a

(i)1,1, a

(i)2,1, . . . , a

(i)d,d

)′

=

1...

1

x(1)i,1 · · · x

(1)i,d

......

x(η)i,1 · · · x

(η)i,d

(

x(1)i,1

)2x

(1)i,2x

(1)i,1 · · ·

(

x(1)i,d

)2

......

...(

x(η)i,1

)2x

(η)i,2 x

(η)i,1 · · ·

(

x(η)i,d

)2

−1

v(i)1...

v(i)η

≡(

M (i))−1

v(i)(t) (3.3.2)

assuming that the matrix M (i) is nonsingular.

Now let us consider the effect of the operator L on v(i). Denoting now the ith

grid point by x(i), we note that

∂v(i)

∂xj(x(i)) = a

(i)j,0 +

d∑

k=1

a(i)j,k(1 + δj,k)x

(i)k

∂2v(i)

∂xj∂xk(x(i)) = a

(i)j,k(1 + δj,k)

where δj,k is the Kronecker delta function. Hence the effect is

(Lv(i))(x(i)) =

α0,0v(i) +

d∑

j=1

αj,0∂v(i)

∂xj+

d∑

j=1

j∑

k=1

αj,k∂2v(i)

∂xj∂xk

(x(i))


= α0,0

a(i)0,0 +

d∑

j=1

a(i)j,0x

(i)j +

d∑

j=1

j∑

k=1

a(i)j,kx

(i)j x

(i)k

+d∑

j=1

αj,0

a(i)j,0 +

d∑

k=1

a(i)j,k(1 + δj,k)x

(i)k

+

d∑

j=1

j∑

k=1

αj,k

a(i)j,k(1 + δj,k)

= a(i)0,0 α0,0 +

d∑

j=1

a(i)j,0

α0,0x(i)j + αj,0

+

d∑

j=1

j∑

k=1

a(i)j,k

α0,0x(i)j x

(i)k + (1 + δj,k)αj,k +

(

αk,0x(i)j + αj,0x

(i)k

)

or in matrix form

(Lv(i))(x(i)) =

α0,0

α1,0 + α0,0x(i)1

...

αd,0 + α0,0x(i)d

α0,0

(

x(i)1

)2+ 2α1,0x

(i)1 + 2α1,1

(

α0,0x(i)2 x

(i)1 + α2,1

)

+(

α1,0x(i)2 + α2,0x

(i)1

)

...

α0,0

(

x(i)d

)2+ 2αd,d + 2αd,0x

(i)d

′

a(i)0,0

a(i)1,0...

a(i)d,0

a(i)1,1

a(i)2,1...

a(i)d,d

= β(α, x(i))′a(i)(t)

and substituting for a and evaluating at x(i) we have

(Lv(i))(x(i)) = β(α, x(i))′(

M (i))−1

v(i)(t)

= A(i)v(i)(t) (3.3.3)

where A(i) is a row vector of length η which we can think of as containing the

elements of the ith row of some matrix A which defines the constrained system of

ordinary differential equations (3.2.4).

3.3. Methodology 47

In order to unify these n equations into the form of Equation 3.2.4, introduce

an operator SX : Rη → Rn which stretches and rearranges the η-vector v(i) so that

it becomes an n-vector with the entries placed in positions corresponding to the

nearest neighbours of x(i).

SX (x, i) ,

(

0, . . . , 0, x(j1), 0, . . . , 0, x(j2), 0, . . . , 0, x(jη), 0, . . . , 0)

where x(jk) is in the Nk,X (i)th position (in particular x(i) is in the ith position).

Now let

A ,

SX (A(1), 1)...

SX (A(n), n)

. (3.3.4)

3.3.2 Weighted least squares

As a simple extension to the above one may consider a least squares regression us-

ing ξ > η nearest neighbours. As a further extension one may consider weighting

the points in the regression according to their distance from xi.

Letting M (i) be the matrix in (3.3.2), but now with ξ rows. Furthermore let Λ

denote a diagonal weighting matrix, yet to be specified. The least squares criterion

then gives

a(i)(t) =(

M (i)′Λ2M (i))−1

M (i)′Λv(i)(t). (3.3.5)

Now, as in (3.3.3), we have the approximation

A(i) , β(α, x(i))′(

M (i)′Λ2M (i))−1

M (i)′Λ (3.3.6)

and we define A from the A(i) as in (3.3.4).

3.3.3 Time stepping

We can now discretise time using a θ-method. Let tk = kTK for some δt, k ∈

0, . . . ,K and θ ∈ [0, 1], which can be thought of as the implicitness.

In the unconstrained case we form the finite difference equation

vk+1 − vk

δt+ (1 − θ)Avk+1 + θAvk = 0


where vk ≡ v(tk). Then the set of equations we have to solve at each time step is

(I + δt(1 − θ)A) vk+1 = (I − δtθA) vk

where the initial conditions are given by vK = ψ(x, T ).

In the constrained case, we obtain the complementarity problem

(I + (1 − θ)Aδt) vk+1 − (I − θAδt) vk ≤ 0

vk − ψ ≥ 0

((I + (1 − θ)Aδt) vk+1 − (I − θAδt) vk)′ (vk − ψ) = 0

(3.3.7)

which can be solved using Cryer’s PSOR [25] or linear programming, for example.

3.3.4 Stability

The stability of the time stepping algorithm depends crucially on the eigenvalues

of the matrix A. In particular, the stability of the time stepping method in the

unconstrained case requires that real eigenvalues of A must be nonpositive. The

mapping of eigenvalues from the matrix A to the time stepping matrix is shown in

Figure 3.3.1.

Algebraically, stability can be guaranteed through the diagonal dominance con-

dition

|aii| ≥∑

j 6=i

|aij | (3.3.8)

where aii ≤ 0 for all i. This condition ensures the real parts of the eigenvalues of

A are negative, a direct consequence of the Gershgorin disc theorem. Since row

sums are zero, this also implies that off-diagonal entries must be nonnegative. The

condition (3.3.8) is not necessary for stability however.

3.4 Application to regular grid

Before applying the discretisation method in its generality, we would first like to

investigate its behaviour on regular grids. In particular it is of interest to compare

the irregular grid method to standard finite difference methods. In the following

analysis we consider only internal grid elements, and not boundary elements.

3.4. Application to regular grid 49

−1/δ t

1

1

(a) Explicit scheme

1

1

(b) Crank-Nicolson scheme

1

1

1/δ t

(c) Implicit scheme

Figure 3.3.1: Mapping of eigenvalues from generator matrix A to time stepping

matrix M in the explicit, Crank-Nicolson and implicit schemes. The shaded areas

correspond to stable eigenvalues.


3.4.1 One dimension

The standard finite difference method on a regular grid approximates the first and

second derivatives as

∂v

∂x(xi) '

vi+1 − vi−1

2δx,

∂2v

∂x2(xi) '

vi+1 − 2vi + vi−1

δx2(3.4.1)

thus leading to the standard finite difference matrix A which has nonzero compo-

nents in row i of Ai where

Ai =(α11

δ2−α10

2δ, α00 − 2

α11

δ2,α11

δ2+α10

2δ

)

. (3.4.2)

This implies the approximation

(Lv) (xi) ' Ai

vi−1

vi

vi+1

. (3.4.3)

Consider now the irregular grid approximation introduced in the previous sec-

tion. Since we are using a regular grid, let us choose our x to be multiples of some

δ and let xi = iδ for all i. Making use of three nearest neighbours, we then have

M (i) =

1 (i− 1)δ (i− 1)2δ2

1 iδ i2δ2

1 (i+ 1)δ (i+ 1)2δ2

(3.4.4)

and thus our approximation to L at xi is

A(i) = β(α, x(i))′(

M (i))−1

=

α00

α10 + α00iδ

α00i2δ2 + 2α10iδ + 2α11

′

12 i(i+ 1) 1 − i2 1

2 i(i− 1)

− 12δ (2i + 1) 1

2δ i12δ (1 − 2i)

12δ2 − 1

δ21

2δ2

=(α11

δ2−α10

2δ, α00 − 2

α11

δ2,α11

δ2+α10

2δ

)

.

Hence we see that for a regular grid in one dimension the method of the previous

section is equivalent to the standard finite difference method.


7 5 9

2 1 3

6 4 8

Table 3.4.1: Assignment of indices of v to neighbours.

3.4.2 Two dimensions

We now compare the irregular grid method to the standard finite difference method

on a regular grid in two dimensions. For the finite difference scheme, in addition

to the derivative approximations given above for one dimension we introduce the

standard cross-derivative approximation (see for example Wilmott [73])

∂2v

∂x1∂x2(xi, xj) '

vi+1,j+1 − vi−1,j+1 − vi+1,j−1 + vi−1,j−1

4δx2. (3.4.5)

In this case the ith row of the approximation matrix is

Ai = (α00 −2(α11 + α22)

δ2,−

α10

2δ+α11

δ2,α10

2δ+α11

δ2, . . .

−α20

2δ+α22

δ2,α20

2δ+α22

δ2,α12

4δ2,−

α12

4δ2,−

α12

4δ2,α12

4δ2)

implying the approximation

(Lv) (xi) ' Ai

vi,j

vi−1,j

vi+1,j

vi,j−1

vi,j+1

vi−1,j−1

vi−1,j+1

vi+1,j−1

vi+1,j+1

. (3.4.6)

Pictorially, we present in Table 3.4.1 the way in which entries in the vector v

correspond to the points around vi,j .


Nine neighbours

In order to compare the finite difference scheme to the irregular grid method, we

consider nine nearest neighbours in the approximation. We thus use the least

squares approach and denote xi,j = (iδ, jδ). The resulting approximation is

A(i) =

59α00 −

2(α11 + α22)

3δ2

29α00 +

2α11 − α10δ − 4α22

6δ2

29α00 +

2α11 + α10δ − 4α22

6δ2

29α00 +

2α22 − α20δ − 4α11

6δ2

29α00 +

2α22 + α20δ − 4α11

6δ2

−19α00 +

4(α11 + α22) + 2(−α20 − α10)δ + 3α12

12δ2

−19α00 +

4(α11 + α22) + 2(α20 − α10)δ − 3α12

12δ2

−19α00 +

4(α11 + α22) + 2(−α20 + α10)δ − 3α12

12δ2

−19α00 +

4(α11 + α22) + 2(α20 + α10)δ + 3α12

12δ2

′

. (3.4.7)

One can view this approximation as a weighted finite difference method, with

the weightings being given in Tables 3.4.2–3.4.7. Note that the only case in which

the weights are the same is for α12; in all other cases the weights for the nine

neighbour irregular grid scheme have less concentrated weights than in the finite

difference method.

0 0 0

0 1 0

0 0 0

1

9

-1 2 -1

2 5 2

-1 2 -1

Table 3.4.2: Weights for α00, for finite difference and irregular grid, respectively.


1

2δ

0 0 0

-1 0 1

0 0 0

1

6δ

-1 0 1

-1 0 1

-1 0 1


1

2δ

0 1 0

0 0 0

0 -1 0

1

6δ

1 1 1

0 0 0

-1 -1 -1


Six neighbours

Alternatively if we use six nearest neighbours, which is the minimum required to

find a quadratic interpolant, we must make a choice between the diagonally located

neighbours. Focusing on the points xi,j ,xi,j−1,xi,j+1,xi−1,j ,xi+1,j and xi+1,j+1

we find that

A(i) =

α00 −2(α11 + α22) − α12

δ2

−α10

2δ+α11 − α12

δ2α10

2δ+α11

δ2

−α20

2δ+α22 − α12

δ2α20

2δ+α22

δ2α12

δ2

′

. (3.4.8)

We thus can find no equivalence between the irregular grid method and the

standard finite difference method on a two dimensional regular grid, in contrast

to the one dimensional case. One can however see the irregular grid method as a

modified finite difference scheme in which different weights are used in the finite

difference approximations.

We shall see in the Section 3.5 that the local quadratic approximation method


1

δ2

0 0 0

1 -2 1

0 0 0

1

3δ2

1 -2 1

1 -2 1

1 -2 1


1

δ2

0 1 0

0 -2 0

0 1 0

1

3δ2

1 1 1

-2 -2 -2

1 1 1


leads to a stable scheme when six neighbours are used as in (3.4.8), but an unstable

scheme when nine neighbours are used as in (3.4.7).

3.5 Experimental results

In order to solve the complementarity problem related to the American option pric-

ing problem, we must find a stable and convergent method for time stepping. As

outlined in Section 3.3.4, a necessary condition for obtaining a stable time stepping

matrix is that the real eigenvalues of A are nonpositive.

Having ascertained that A will lead to a stable time stepping scheme, it also

remains to check the convergence conditions for the LCP solution method.

3.5.1 Grids on the unit cube in R2

We first consider grids on the region Ω = [0, 1]2 ⊂ R2. This is a natural place to

start since we know that finite difference schemes using regular grids lead to stable

A matrices in this case. Boundaries are allowed to be neighbours in this setting.

We present in Figures 3.5.1–3.5.7 point sets of size approximately 500 and the

eigenvalues of the corresponding Amatrix obtained using the irregular grid method

(plotted in the complex plane).


−4 −3 −2 −1 0 1 2−4

−3

−2

−1

0

1

2

−20 −15 −10 −5 0−8

−6

−4

−2

0

2

4

6

8x 10

−12

Figure 3.5.1: Points and eigenvalues of A for regular grid with 529 interior points

and using 6 neighbours - stable. Note that the vertical scale in the eigenvalue plot

is close to zero.

−4 −3 −2 −1 0 1 2−4

−3

−2

−1

0

1

2

−10 −5 0 5 10−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Figure 3.5.2: Points and eigenvalues of A for regular grid with 529 interior points

and using 9 neighbours - unstable.

−4 −3 −2 −1 0 1 2−4

−3

−2

−1

0

1

2

−25 −20 −15 −10 −5 0−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 3.5.3: Points and eigenvalues of A for triangular grid with 546 interior

points and using 6 neighbours - stable.


−4 −3 −2 −1 0 1 2−4

−3

−2

−1

0

1

2

−20 −15 −10 −5 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 3.5.4: Points and eigenvalues of A for hexagonal grid with 512 interior


−4 −3 −2 −1 0 1 2−4

−3

−2

−1

0

1

2

−2000 −1000 0 1000 2000 3000−80

−60

−40

−20

0

20

40

60

80

Figure 3.5.5: Points and eigenvalues of A for uniform pseudo-random grid with

500 interior points and using 6 neighbours - unstable.

−4 −3 −2 −1 0 1 2−4

−3

−2

−1

0

1

2

−5000 0 5000 10000 15000 20000−8

−6

−4

−2

0

2

4

6

8

Figure 3.5.6: Points and eigenvalues of A for Sobol’ grid with 500 interior points

and using 6 neighbours - unstable.


1

4δ2

-1 0 1

0 0 0

1 0 -1

1

4δ2

-1 0 1

0 0 0

1 0 -1


−4 −3 −2 −1 0 1 2−4

−3

−2

−1

0

1

2

−25 −20 −15 −10 −5 0−1.5

−1

−0.5

0

0.5

1

1.5

Figure 3.5.7: Points and eigenvalues of A for low distortion grid with 500 interior


In the case of the Sobol’ grid one can in practise observe which points are caus-

ing instability by examining the eigenvector corresponding to the eigenvalues with

positive real part. The instability can often be resolved by changing the neighbour

configuration, in particular so that neighbours are well-distributed about the point.

No systematic method was found to perform this stabilisation however.

Note that the least squares scheme using nine neighbours was not stable, de-

spite the fact that it uses the same points as in the regular finite difference scheme.

It was also found that none of the grids considered above leads to a stable A when

considering local least squares fits over 7 neighbours.

3.5.2 Normally distributed grids in R2

Grids constructed on the unit cube are not optimal for the application under consid-

eration. The main reason for this is that they are not representative of the regions

of space that are likely to be visited by the stochastic process introduced in (3.2.1).

A further problem with using a grid on the unit cube is that neither Dirichlet


−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

−80 −60 −40 −20 0 20−6

−4

−2

0

2

4

6

Figure 3.5.8: Points and eigenvalues of A for low distortion normal grid with 500

points and using r = ∞ and 6 neighbours - unstable, max(<(λ)) = 2.54.

nor Neumann boundary conditions are known. Approximate conditions may be

specified, thus adding an extra source of error to the computed solution.

A natural grid choice for our problem would be one that is related to the pro-

cess, and in this case a normally distributed grid seems appropriate. This also

alleviates the second problem mentioned, in that a normally distributed grid cov-

ers Rd asymptotically, and so boundary conditions have a vanishing effect on the

solution.

Given the previous results of grids on the unit cube, we choose to focus on low

distortion grids in the following. We generate a low distortion grid with respect

to the standard normal density in R2, and apply the irregular grid method to it to

obtain the matrix A. We then examine the eigenvalues of A.

Since there are no natural boundary points when dealing with normally dis-

tributed grids, we choose a radius outside which all points are considered to be

boundary points. In particular we consider the radii r = ∞, 3.0, 2.5 and 2.0. In

general one expects the eigenvalues to be different for different choices of r, and

in particular that A should only be stable for smaller choices of r.

The results are presented in Figures 3.5.8 – 3.5.11. The maximum real parts

of the eigenvalues are given in the captions. In this case the transition to stability

occurs when r is between 3.0 and 2.5.


−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

−80 −60 −40 −20 0 20−6

−4

−2

0

2

4

6


points and using r = 3.0 and 6 neighbours - unstable, max(<(λ)) = 2.51.

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

−80 −60 −40 −20 0−6

−4

−2

0

2

4

6


points and using r = 2.5 and 6 neighbours - stable, max(<(λ)) = 0.

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

−80 −60 −40 −20 0−6

−4

−2

0

2

4

6


points and using r = 2.0 and 6 neighbours - stable, max(<(λ)) = 0.


3.6 Conclusions

We presented a method for approximating a differential operator on an irregular

grid. The method uses local polynomial interpolants to construct derivative approx-

imations. We analysed the stability of the operator approximation using different

grid and boundary configurations.

In one dimension, we showed that the method is equivalent to the standard

finite difference method.

Our main finding in two dimensions was that grids with a regular local structure

are more likely to lead to stable approximations. Thus square, triangular and hexag-

onal grids lead to stable approximations, but pseudo-random and quasi-random

grids did not. Low distortion grids as used in Bally and Pages [3] were also found

to lead to stable approximations. We were able to induce stability in the case of a

low discrepancy grid by altering neighbour configurations so that the neighbours

were more uniformly distributed in an angular sense. For a low distortion grid

adapted to the normal distribution, we found that the approximations constructed

were unstable when the boundary radius was too large, but stable for smaller radii.

Summarised, this study indicates that instabilities in the approximation are a

consequence of the local roughness of points in the grid, and of boundary effects.

The hurdle in extending this work to higher dimensions is stability, in particular

more research is required either in the direction of determining sufficient conditions

for stability on arbitrary grids or towards modifications of the approximation me-

thod. Such conditions are provided in Chapter 6; however the satisfaction of these

conditions has not yet been investigated for the method presented in this paper.

The literature on mesh-free methods provides one solution in the form of the

moving least squares method, where one attempts to integrate a particular inter-

polant, as opposed to working with derivatives of the interpolants. The moving

least squares method seems to be less susceptible to instabilities than the present

method; this is investigated further in Chapter 5.

Chapter 4

A Method Using LocalConsistency Conditions

4.1 Introduction

The pricing of American options is a problem that has remained inaccessible to

closed form solution. It was also long assumed to be inaccessible to Monte Carlo

techniques, but Tilley quashed this belief in his 1993 paper [70]. Simulation tech-

niques are of particular importance for higher dimensional problems where con-

ventional discretisation methods become intractable.

Methods for solving American and Bermudan option pricing problems have

become increasingly important with the widespread use of options and the de-

velopment of more and more complex contracts. Examples of potentially high-

dimensional options include basket options, swaptions and real options. We con-

sider “high-dimensional” problems to be those where the number of stochastic

factors is at least three or four, and thus conventional grid techniques become un-

manageable.

Much progress has been seen in the past decade in the area of Monte Carlo

techniques, through the work of Barraquand and Martineau [4], Broadie and Glass-

erman [19] and more recently Longstaff and Schwartz [50], Tsitsiklis and Van Roy

[71], Rogers [64], Haugh and Kogan [37], Boyle et al. [14], and through the me-

thod proposed in Chapter 2 (also published as Berridge and Schumacher [8, 7, 9]).

Most techniques proposed have centred around path generations of the process.

61

62 Chapter 4. Local Consistency Conditions

This has the advantage that the points sampled are well adapted to the process, but

the disadvantage that it is difficult to determine the expected value of continuation

at each point. It is important to know the latter in order to make a stopping decision,

and thus determine the early exercise premium.

The last method in the above list is the only one to consider a constant sam-

pling of the state space over time. Since the method centres around an approxi-

mating Markov chain, it is simple to estimate continuation values on the grid using

an appropriate Markov transition matrix. This method is thus more like a finite

difference method, as opposed to the methods in [19, 50, 71, 14] which are more

tree-like.

An important advantage of the irregular grid method proposed here is that the

number of tuning parameters is small. Furthermore, convergence requires increas-

ing only the number of grid points and the number of time steps, as with finite

difference methods. In particular the method does not involve approximation of

the value function or exercise region by basis functions.

We also note that using a constant grid allows implicit solutions to be easily

obtained; for finite difference techniques this represents an increase in convergence

speed from δt to δt2 when considering European problems.

We proceed along the lines of Chapters 2 and 3 in that we approximate the

value function on an irregular grid. We use a stable and more tractable method

however for approximating the transition probabilities; instead of taking a root of

a transition matrix, we directly construct the transition probabilities using local

consistency conditions presented in Kushner and Dupuis [46] in the parabolic case

and similar conditions to construct the infinitesimal generator in the elliptic case.

This allows us to use much larger grids, and thus obtain more accurate solutions.

Using the root method in Chapter 2 the grid size was limited to 3000 on a

desktop computer, and averaging was needed over several experiments to obtain

accurate solutions. We can now deal with grid sizes in the hundreds of thousands,

and solutions from a single experiment are of sufficient accuracy that randomisa-

tion is no longer required.

This chapter continues in Section 4.2 with a formulation of the problem of in-

terest. Section 4.3 presents the proposed methodology, refinements are presented in

Section 4.4 and experiments are carried out in Section 4.5. Section 4.6 concludes.

4.2. Formulation 63

4.2 Formulation

4.2.1 The market

As in Chapter 2, we consider a complete and arbitrage-free market described by

state variable X(s) ∈ Rd for s ∈ [t, T ] which follows a Markov diffusion process

dX(s) = µ(X(s), s)ds+ σ(X(s), s)dW (s) (4.2.1)

with initial condition X(t) = xt, and a derivative product on X(s) with exercise

value ψ(X(s), s) at time s and value V (s) = v(X(s), s) for some pricing function

v(x, s). The process V (s) satisfies

dV (s) = µV (X(s), s)ds + σV (X(s), s)dW (s) (4.2.2)

where µV and σV can be expressed in terms of µ and σ by means of Ito’s lemma.

The terminal value is given by V (·, T ) = ψ(·, T ), and intermediate values satisfy

V (·, s) ≥ ψ(·, s), s ∈ [t, T ]. It is assumed that µ(x, s) and σ(x, s) satisfy suitable

regularity conditions, such that the change of measure implied by (4.2.2) is well-

defined.

In such a market there exists a unique equivalent martingale measure under

which all price processes are martingales. The risk-neutral process in this case is

given by

dX(s) = µRN (X(s), s)ds+ σ(X(s), s)dW (s) (4.2.3)

where µRN is the risk-neutral drift.

Our objective is to determine the current value V (X(t), t) of the derivative

product and the accompanying adapted exercise and hedging strategies τ and H:

τ : Rd × [t, T ] → 0, 1 (4.2.4)

H : Rd × [t, T ] → Rd. (4.2.5)

Supposing that one has an estimate V (t) of the derivative price, it is often

important to specify an exercise rule τ or a hedging strategy H in order for the

buyer or seller, respectively, to be able to realise the estimated price.


4.2.2 Pricing

The primal formulation

The value of the derivative product is formulated in the primal problem as a supre-

mum over stopping times

v(xt, t) = supτ∈T

EQxt

(

e−r(τ−t)ψ(X(τ), τ))

(4.2.6)

where T is the set of stopping times on [t, T ] with respect to the natural filtration,

the expectation is taken with respect to the risk-neutral measure Q, and the initial

value is X(t) = xt.

The dual formulation

The dual formulation (see Rogers [64] or Haugh and Kogan [37]) forms a price by

minimising the cost of the hedging strategy over martingales. Theorem 1 of [64]

implies that the price is given by

v(xt, t) = infM∈H1

0

EQxt

[

sups∈[t,T ]

(

e−r(s−t)ψ(X(s), s) −M(s))

]

(4.2.7)

where H10 is the space of martingales with M(0) = 0 and sups∈[t,T ] |M(s)| ∈ L1.

The infimum is attained at a certain martingale M = M ∗.

The variational inequality formulation

Formulating the problem as a variational inequality invites implications from the

large number of results that have been developed in this field, for example the work

of Glowinski et al. [35]. Jaillet et al. [42] applied this approach to the analysis of

American option pricing.

One must first define an elliptic operator L giving the diffusion of the process.

This is given by

L = 12 trσσ′

∂2

∂x2+ µRN

∂

∂x− r (4.2.8)

where r is the risk-free rate.

One must also specify a function space in which to work. Briefly one defines

an inner product 〈·, ·〉 and a bilinear form a(·, ·) on the Hilbert space H 1 satisfying

a(v, u) = 〈u,Lv〉 . (4.2.9)

4.3. Methodology 65

The equivalent variational inequality formulation is then to find v(x, t) such

that

v(x, s) − ψ(x, s) ≥ 0

u ≥ ψ a.e. ⇒ a(v, u− v) −⟨

u− v, ∂v∂t

⟩

≥ 0 a.e. [t, T ](4.2.10)


The complementarity formulation

The variational inequality formulation is not directly amenable to computation. For

this reason it is convenient to reformulate it as a complementarity problem. Let L

be the related diffusion operator; then the option value is found by solving the

complementarity problem

∂v∂t + Lv ≤ 0

v − ψ ≥ 0(

∂v∂t + Lv

)

(v − ψ) = 0

(4.2.11)


Such a problem can be solved using standard PDE discretisation techniques,

with some modifications to account for the inequalities.

4.2.3 Consequences

In solving the pricing problem we divide the time-state space into two complemen-

tary regions: the continuation region where it is optimal to hold the option and the

stopping region where it is optimal to exercise. In the continuation region the first

line of (4.2.11) is active and the stopping rule says not to exercise. In the stopping

region the second line of (4.2.11) is active and the stopping rule says to exercise.

In all formulations presented, high dimensionality poses a practical problem

since functional approximation in a high-dimensional space is called for.

4.3 Methodology

The basic methodology presented is similar to that of Chapter 2, with the excep-

tion of the manner in which the transition matrix is constructed. This is now done


−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

(a) Random

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

(b) Sobol’

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

(c) Low Distortion

Figure 4.3.1: Grids with 500 points adapted to the normal density.

using the local consistency conditions presented in Kushner and Dupuis [46], and

a modification of these conditions is used to find an approximation to the infinites-

imal generator. These conditions ensure that the approximating Markov chain has

a local mean and variance that match those of the continuous process.

4.3.1 Irregular grid

We first briefly review the irregular grid methodology presented in Chapter 2. We

define an irregular grid to be a representative sampling of the state space

X = x1, . . . , xn ⊂ Rd. (4.3.1)

The method of sampling is to be specified at a later stage, but one can think of it

as a low discrepancy or low distortion set (see for example Bally and Pages [60])

which is dense in the entire state space as n→ ∞.

Examples of possible grids in two dimensions are presented in Figure 4.3.1.

As in the case of Monte Carlo integration, it is expected that low discrepancy (e.g.

Sobol’) and low distortion grids will lead to faster convergence than random grids.

For results regarding integration see Evans and Swartz [28] and Pages [60].

In order to simplify the analysis we now make the assumption that the risk-

neutral process is a d-dimensional time homogeneous diffusion process

dX(s) = µRN (X(s))ds +R(X(s))dW (s) (4.3.2)

where R′R is the Cholesky decomposition of the state-dependent covariance ma-

trix Σ(X(s)) and X and W are of the same length d. This assumption is not

necessary for the method to work; it merely simplifies some aspects and allows for

a clearer exposition.

4.3. Methodology 67

4.3.2 Approximation of Markov chain

We consider approximating the risk-neutral process (4.3.2) using a discrete state,

discrete time Markov chain where the states are exactly the points in our irregular

grid X and the time step is δt.

The Markov transition matrix P is constructed in such a way as to satisfy the

local consistency conditions given in Kushner and Dupuis [46]. We require1 for

each state i = 1, . . . , n

Σ(xi)δt =∑n

j=1(xj − xi − µRN (xi)δt)(xj − xi − µRN (xi)δt)′pi,j

µRN (xi)δt =∑n

j=1(xj − xi)pi,j

1 =∑n

j=1 pi,j

pi,j ≥ 0(4.3.3)

where pi,j is the (i, j)th entry of P .

One must solve for each state i a feasibility problem over the pi,j . The number

of equality constraints in the problem is given by ηd + 1 where

ηd =1

2d(d+ 3) (4.3.4)

and the number of variables is n. In the problems we consider, ηd is much smaller

than n.

In practise one can impose the extra condition that the transitions should only

be allowed to close neighbours of each point. Computationally this means that we

only need to consider a small number of transitions k where ηd +1 < k n, thus

dramatically reducing the complexity of the problem.

It is also useful to specify a linear objective function to optimise the proximity

of transitions. That is, to satisfy the local consistency conditions using points as

close as possible to the mean. The linear objective function, to be minimised,

should have a coefficient relating to point xj which is an increasing function of the

distance ||xi − xj||. Let us denote the objective function by fi · pi where pi is the

ith row of P .

1The formulation in [46] is more general in that it allows o(δt) terms to be added on the RHS of

the first two conditions.


We thus pose for each point xi a linear program min fi · pi subject to (4.3.3).

In experiments we found that a convenient specification for f is fj = k3 where xj

is the kth nearest neighbour of xi + µRN δt.

We note that the solution to the linear program will in general be a corner

solution using as many zero variables as possible; the number of nonzero transition

probabilities per point is the minimum number, ηd + 1. This is a consequence

of Corollary 7.11 in Schrijver [66], and of the fact that the constraint matrix has

ηd + 1 rows. Note that the points with positive weights are not necessarily the

ηd +1 nearest neighbours of xi +µRNδt, since these may not satisfy the feasibility

conditions; the points form rather the closest possible feasible set (with respect to

the objective function).

4.3.3 Approximation of infinitesimal generator

Rather than approximating transition probabilities, one may attempt to approxi-

mate the infinitesimal generator directly. This amounts to constructing a discrete

space, continuous time approximation to the problem.

Constructing an approximation to the infinitesimal generator allows quick re-

construction of transition probabilities for arbitrary time steps δt, or for scaling the

effect of the diffusion operator, through a first order approximation. Consequently

this method is preferred over that of Section 4.3.2, provided we do not have a large

state-dependent drift. We assume the latter in this section.

In the case of a non-state dependent drift, we refer the reader to Section 4.4.5

where we introduce a simple transformation of the continuous process to eliminate

a risk-neutral drift that depends deterministically on time.

We start with the problem (4.3.3), and define

ai,j =1

δt(pi,j(δt) − δij) (4.3.5)

where δij is the Kronecker delta. As δt → 0 in (4.3.5) we obtain elements of the

infinitesimal generator matrix A.

Substituting (4.3.5) into (4.3.3) and letting δt → 0 yields the new feasibility

problemΣ(xi) =

∑

j 6=i(xj − xi)(xj − xi)′ai,j

µRN (xi) =∑

j 6=i(xj − xi)ai,j

ai,j ≥ 0

(4.3.6)

4.3. Methodology 69

and ai,i = −∑n

j 6=i ai,j . Note that (4.3.6) now contains only ηd equality constraints,

one less than (4.3.3).

The same considerations as in Section 4.3.2 are also applied in this case. We

solve for each point xi a linear program min f · ai subject to (4.3.6) and ai,j ≥ 0

where ai is the ith row of A with the diagonal entry omitted. Following from the

observation at the end of Section 4.3.2, we again expect a maximum of ηd + 1

nonzero entries per row of A.

We note that when a large drift term is present, one may be able to satisfy the

local consistency conditions (4.3.6), but this may require using points xj which

are nonlocal to xi. This method differs from the usual method of lines in that

here we produce a stable system before checking for localness of the neighbours,

whereas in the usual method one selects the neighbours a priori before building

the equations and finally considering stability (see for example Hundsdorfer and

Verwer [41]).

4.3.4 Time stepping

Given a transition matrix P , corresponding to time step δt, the option pricing prob-

lem can be solved using dynamic programming on the discretised Markov chain.

Namely, one solves the problem

v(T ) = ψ

v(tk) = max(

ψ, e−rδtPv(tk+1))

for tk = kδt and k = K−1, . . . , 0 whereK is the number of time steps considered,

v is a vector of values at grid points and ψ is a vector of payoffs at grid points,

here assumed to be constant over time. The resulting solution v(x, 0) is an exact

solution to the approximating Markov chain.

Given the infinitesimal generator A, one can form a first order approximation

to the transition matrix P ' I + Aδt and proceed as above. Alternatively, it is

possible to solve the problem to a higher order using the matrix exponential

v(T ) = ψ

v(tk) = max(

ψ, e−rδteAδtv(tk+1))

for k = K − 1, . . . , 0. Since A is sparse, the effect of the matrix exponential can


be calculated efficiently using Krylov subspace methods, see for example Druskin

and Knizhnerman [27] or Hochbruck and Lubich [38].

The above time stepping methods are suitable for Bermudan pricing problems

with δt being the period between exercise possibilities. We expect convergence to

the Bermudan solution as n → ∞, and convergence to the American solution as

δt→ 0.

When considering a truly American problem, it is useful to consider Crank-

Nicolson and implicit solutions. In particular the Crank-Nicolson method is known

to converge at a rate δt2 for the European problem as opposed to δt for the explicit

and implicit methods, and implicit methods are known to be unconditionally stable

for solving sequences of LCPs (see Chapter 6 and Glowinski et al. [35]).

The Crank-Nicolson method corresponding to the truly American problem is

the following system with θ = 12

v(T ) = ψ (4.3.7)

0 ≤ (v(tk) − ψ) ⊥(

e−θAδtv(tk) − e−rδte(1−θ)Aδtv(tk+1))

≥ 0

for k = K−1, . . . , 0. The second line is a linear complementarity problem (LCP).

There are many methods available for solving LCPs, including the projected suc-

cessive overrelaxation (PSOR) method proposed in Cryer [25]. Another possible

candidate is linear programming, which is used for example by Dempster and Hut-

ton [26] to solve the one-dimensional American option pricing problem.

4.3.5 Summary of the algorithm

We present a concise statement of the proposed algorithm as Algorithm 4.3.1. The

generation of the matrices A can be done in advance for a given grid X , with

obvious changes to the algorithm.

4.4 Fine tuning and extensions

We now mention some implementation issues and refinements of the method. These

issues are not essential to the method, but may improve performance and allow

quicker execution for a given required accuracy.

4.4. Fine tuning and extensions 71

Algorithm 4.3.1 Proposed algorithm for solving high-dimensional American op-

tion pricing problems.Choose the grid size n

Generate a QMC grid X

Compute the generator matrix A

Choose the time step δt > 0 and implicitness θ ∈ [0, 1]

Solve the linear complementarity problems (4.3.7)

4.4.1 Grid specification

In the presentation so far, we have taken the grid X to be given; we now consider

ways one might specify the grid.

Taking inspiration from the literature on MC and QMC integration, we first

suggest that the grid be constructed using low discrepancy (Niederreiter [58]) or

low distortion (Pages [60]) methods. Just as in the regular grid case, we expect the

error to be related to the separation of grid points, more specifically the separation

of grid points having positive weights in the generator matrix.

Importance sampling considerations tell us that the most efficient grid density

is given by the density of the process itself. Given our suggestion of a constant grid

(for efficiency reasons), we cannot provide the most efficient importance sampling

at all times. However, given the restriction to a constant grid, we can still provide

an acceptable importance sampling.

As outlined in Evans and Swartz [28], the rate of convergence for importance

sampling of normal densities using normal importance sampling functions is most

damaged when the variance of the importance sampling function is less than that

of the true density. Conversely, convergence rates are not greatly affected when

the variance of the importance sampling function is greater than that of the true

density. The situation we should try to avoid is that the process has a significant

probability of lying in the “tails” of the grid density.

A further consideration is the minimisation of boundary effects on the solution.

This suggests that the grid covariance should be larger than the covariance of the

process.

In Chapter 2, where a root method was used to construct transition probabil-

ities, and the process considered was a five-dimensional Brownian motion with

drift, a grid covariance of 1.5 times the process covariance at expiry was found to


−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

x1

x 2

Figure 4.4.1: Interior points (small) and boundary points (large) on a normal low

distortion grid for d = 2, n = 500.

give the best convergence rate when tested against grids with covariances of 1.0

and 2.0 times the covariance at expiry.

4.4.2 Boundary region and boundary conditions

It is clear that (4.3.3) and (4.3.6) may be infeasible for some i. In such a case we say

that xi is an implied boundary point, otherwise it is an implied interior point. Given

nondegenerate Σ and a well-adapted grid, one expects that the implied boundary

points will indeed lie at the extremities of the grid, and the implied interior points

away from the extremities.

One may specify appropriate boundary conditions in this region to reflect the

behaviour of the process. In the experiments we let these points be absorbing,

which is appropriate for value functions having a linear behaviour at the boundary.

One may also apply Dirichlet, Neumann or mixed conditions using neighbours in

the grid.

It would be useful to know a priori which points are likely to be in the implied

boundary, since we would like to avoid trying to solve infeasible linear program-

ming problems. In practise however it is difficult to do this even for simple cases.

A plot of the boundary behaviour for a 500-point low distortion grid in two di-

mensions is given in Figure 4.4.1. Notice in this case that the number of infeasible

points is 21, this being about 4% of the total.

If one assumes a distribution for the neighbours over which (4.3.3) or (4.3.6) is


to be solved, then one can quantify the probability of feasibility. Near the boundary

of the grid, there may be a low density of points on the boundary side, and thus the

probability of feasibility changes.

For example, if our grid consists of n independent standard normal draws, we

can calculate the expected number of grid points in a halfspace away from the

centre of the grid at some radius r. One can then say what the minimum number

of points n is where the expected number of grid points in the halfspace away from

the grid centre at radius r is less than some bound.

Let us set this bound to be 12ηd, where ηd is given in (4.3.4), a very optimistic

bound but useful to illustrate the approximate behaviour of the boundary. Requir-

ing an expected number of 12ηd points in the halfspace away from the center implies

a boundary radius of

r = Φ−1

(

1 −1

2nηd

)

(4.4.1)

where Φ is the cumulative normal distribution function. In order to find the ex-

pected number of boundary points we then note that the squared norm of a standard

normal variable in d dimensions is a chi square random variable with d degrees of

freedom. Thus, if the boundary region is defined by

x : ‖x‖2 ≥ r2

, then the

expected numbers of interior and boundary points are

ENi = nΨ(r2, d) (4.4.2)

ENb = n(

1 − Ψ(r2, d))

, (4.4.3)

respectively, where Ψ is the chi square cumulative distribution function.

Plots of the radius and expected number of boundary points are presented in

Figure 4.4.2 for d = 3, 5, 10 and n up to 300, 000.

Experimentally we find that (4.4.1) underestimates the implied radius for lower

dimensions and overestimates it for higher dimensions (see Section 4.5 for numer-

ical results). The latter is not surprising since one generally requires more than

the minimum number of points ηd to satisfy the feasibility conditions (4.3.3) and

(4.3.6).

Finally we mention that in estimating the boundary, we prefer an underesti-

mate to an overestimate. An overestimate of the boundary may lead us to waste a

considerable amount of computing time trying to solve infeasible linear programs.

An underestimate on the other hand just results in the grid having extra boundary


0.5 1 1.5 2 2.5 3

x 105

3.2

3.4

3.6

3.8

4

4.2

n

r

d=10

d=5

d=3

(a) Radius

0.5 1 1.5 2 2.5 3

x 105

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

n

p d=10

d=5

d=3

(b) Proportion

Figure 4.4.2: Naive prediction for the radius of the boundary and the proportion of

points which are in the boundary region for a standard normal grid.

points. The latter does not add a significant amount of overhead to the method,

the effect being limited to a slight increase in complexity of the nearest neighbour

problem and extra zero rows to the sparse generator matrix.

4.4.3 Parallelism

In the language of computer science, problems (4.3.3) and (4.3.6) are said to be

embarrassingly parallel. This refers to the fact that a speedup linear in the number

of processors can be trivially achieved. For example, having a large number n of

linear programs to solve and m computers, we can reduce the time by a factor 1/m

by solving n/m linear programs per computer, assuming n > m and that the com-

munication time between the computers is negligible. We make use of this point

when conducting the experiments, using a distributed computing environment to

solve the linear programs.

4.4.4 Control variates and Richardson extrapolation

To obtain more accurate solutions we consider variance reduction and extrapolation

techniques.

Variance reduction is already used in the method in that the grids are con-

structed using points designed to cover the state space evenly according to the

process density at expiry. In the current context the idea of control variates is also


very easy to apply since the European solution is usually highly correlated with the

Bermudan and American solutions. Since the European price is easy to determine

to a high degree of accuracy, it constitutes an ideal control variate.

The concept of extrapolation is also useful once we have an idea of how the

error behaves with increasing n. In Section 4.5.5 below, experimental evidence is

given which implies the estimates behave asymptotically as

vn = v + c1nc2/d (4.4.4)

for some constants c1, c2, which may be estimated. Here we assume that the error

is always of the same sign, which may be indicated for example by a monotone

behaviour of the approximations.

4.4.5 Matrix reuse

Given that generating the transition and infinitesimal generator matrices is an ex-

pensive operation compared to the final time stepping procedure, it is of interest to

know under which conditions these matrices can be reused for related problems. It

is clear that a single matrix can be reused for as many different payoff functions as

required; it can also be reused for processes with different risk-neutral drifts and

covariances as follows.

Suppose that a transition or infinitesimal generator matrix has been constructed

for a process with covariance matrix I and zero risk-neutral drift on the grid X . Let

us construct the grid Y where yi = R′xi, R being a Cholesky factor of the covari-

ance matrix Σ. The implied covariance of the transition or infinitesimal generator

matrix on Y is now Σ.

Suppose now that our process has covariance Σ, and constant (nonzero) risk-

neutral drift µ. Consider now the time dependent grid Yk where the subscript

k corresponds to time kδt and yk = x + kµδt. The implied covariance of the

transition or infinitesimal generator matrix remains Σ, but the implied drift is now

µ.

Two simple extensions to the time homogeneous problem are those in which

the risk-neutral drift is deterministically time-dependent and the covariance matrix

is scaled over time,

dX(s) = µRN (s)ds+ α(s)RdW (s). (4.4.5)


The most convenient way to deal with the drift term is to incorporate the drift in

the payoff function. This amounts to the change of variables

X0(s) = X(s) −

∫ s

0µRN (u)du, (4.4.6)

the new process having zero drift

dX0(s) = α(s)RdW (s) (4.4.7)

and the payoff being

ψ0(xi, s) = ψ

(

xi +

∫ s

0µRN (u)du, s

)

. (4.4.8)

The scaled covariance term can be accommodated by manipulating the time step.

By using time step α(s)δt at time s in place of δt, we achieve a covariance of

α(s)2Σ as required.

4.4.6 Grid expansion

Grid expansion relates the size of the grid to the variance of the process. A con-

venient way to generate a grid is to sample the process at expiry; one thus obtains

a grid X that becomes dense in the state space as n → ∞. For a finite n how-

ever one can ask how well the process can be represented on X . For example if

we consider a standard d-dimensional Brownian motion on s ∈ [0, 1], the process

density at expiry is N (0, I). If the implied boundary begins at r < 2 for example,

there is a nonnegligible chance of the discrete Markov process hitting the absorbing

boundary before expiry, thus reducing the accuracy of the solution.

In this case we can set a lower limit r0 for the implied boundary, for example

r0 = 4 for which the process has a negligible chance of hitting the boundary. This

limit can be achieved by expanding the grid; to do this, one scales the grid points

by a factor r0/r and the generator matrix entries by a factor r/r0, thus removing

the boundary effects while preserving local consistency.

The grid expansion factor allows us to make a tradeoff between errors caused

by the boundary and errors related to the discretisation. The higher the factor

applied in the grid expansion, the lower the effect from the boundaries but the

coarser the grid becomes and hence the higher the discretisation error.

4.5. Experiments 77

4.4.7 Partially absorbing boundaries

Infeasibility of points in the boundary region is usually caused by a lack of points

in the halfspace away from the center of the grid. If the grid boundary looks lo-

cally linear, as in a spherical grid, it is possible that the infeasibility is only in this

direction, and not “along” the boundary.

In this case it may be useful to consider partially absorbing boundaries in which

one only tries to satisfy local consistency conditions in the direction tangent to the

boundary. In the case of a normal grid this amounts to requiring a zero variance

along lines through the grid center for points in the boundary layer. This type of

boundary condition has not been employed in the current study.

4.5 Experiments

A major hurdle in testing algorithms for pricing high-dimensional American op-

tions is the difficulty of verifying results. One common method is using out-of-

sample paths to estimate the value of the exercise and hedging strategies implied

by the model. Another, which we use here, is to use benchmark results from a spe-

cial case that can be solved accurately. In the following we introduce benchmark

results and then test the proposed method against those results.

4.5.1 Geometric average options

We choose to focus on geometric average options, since the pricing problem for

these options can be reduced to a one-dimensional problem. The one-dimensional

problems can be solved to a high degree of accuracy, thus providing benchmark

results for the algorithm.

A geometric average put option written on d assets following the risk-neutral

process (4.2.3) has payoff function

ψ(s) =

(

K −(

∏

si

)1/d)+

(4.5.1)

where s is the asset value and K is the strike price of the option. Assuming a com-

plete and arbitrage free market with the log asset prices following a multivariate

Brownian motion with constant covariance Σ, we have a constant risk-neutral drift

µRN = r11 −1

2diag Σ. (4.5.2)


4.5.2 Benchmarks

Using Ito’s lemma with Y = f(X) = X , we find that Y follows the risk-neutral

process

dY (s) =1

d

d∑

i=1

dXi(s) (4.5.3)

= µds+ σdW (s), (4.5.4)

the parameters of the diffusion being given by

µ = r −1

2d

d∑

i=1

σ2i (4.5.5)

σ2 =1

d2

d∑

i=1

d∑

j=1

Rij

2

. (4.5.6)

The option is thus equivalent to a standard put option on an asset with starting value

expX0, strike price K , risk-free rate r and continuous dividend stream

δ =1

2

(

1

d

d∑

i=1

σ2i − σ2

)

. (4.5.7)

In Table 4.5.1 we provide benchmark results for geometric put options written on

up to ten assets, with starting asset values Si = 40 for all i and strike price 40. The

risk-free rate is taken as 0.06, the volatilities σi = 0.2 for all i, and correlations

ρij = 0.25, i 6= j.

4.5.3 Experimental details

Using the methodology proposed in Section 4.3, we conducted experiments to find

the value of the geometric average put options given above.

We used six different grid sizes ranging from 50, 000 to 300, 000, and two

types of grids consisting of normal Sobol’ points and normal low distortion points

with a covariance corresponding to 1.5 times the process covariance at expiry. The

transition matrices were generated using distributed computing software in a Mat-

lab environment. A maximum of 20ηd nearest neighbours were considered when

trying to satisfy the local consistency conditions, where ηd is defined in (4.3.4).

4.5. Experiments 79

d σ2 × 102 δ × 102 European Bermudan American

1 4.000 0.000 2.0664 2.2930 2.3196

2 2.500 0.750 1.5553 1.7557 1.7787

3 2.000 1.000 1.3468 1.5380 1.5597

4 1.750 1.125 1.2318 1.4193 1.4392

5 1.600 1.200 1.1585 1.3421 1.3625

6 1.500 1.250 1.1077 1.2893 1.3094

7 1.429 1.286 1.0703 1.2504 1.2703

8 1.375 1.313 1.0416 1.2207 1.2404

9 1.333 1.333 1.0189 1.1971 1.2167

10 1.300 1.350 1.0004 1.1779 1.1974

Table 4.5.1: Benchmark results for geometric average options in dimensions 1–10.

Also displayed are the variance σ2 and continuous dividend δ for the equivalent

one dimensional problem.

We consider the pricing problem for European options, Bermudan with ten

exercise opportunities and true American where the option can be exercised at

any time up to expiry. For the European and Bermudan problems we used the

Crank-Nicolson method with 100 time steps. For solving the linear systems we

used the conjugate gradients squared (CGS) and generalised minimum residual

(GMRES) methods, the latter being slower but more robust. For the American

problems we used projected successive overrelaxation (PSOR) to solve the linear

complementarity problems, with 1000 time steps. While it is not necessary to use

such a large number of time steps in practise, we wanted to focus on the error

with respect to the space discretisation. Having a small enough δt causes the error

resulting from time discretisation to be negligible in comparison, and thus allows

a more accurate assessment of the error resulting from space discretisation.

4.5.4 Experimental results

We present results in Tables 4.5.2–4.5.4 for prices obtained using normal Sobol’

grids for the Bermudan, American and European cases, respectively. The results

for low distortion grids are presented in Tables 4.5.7–4.5.9.


Tables 4.5.5 and 4.5.6 show the results on normal Sobol’ grids for Bermudan

and American options when the European is used as control variate. Tables 4.5.10

and 4.5.11 show the same for low distortion grids.

Figures 4.5.1 and 4.5.2 present the results graphically for normal Sobol’ grids.

The results for low distortion grids are shown in Figures 4.5.3 and 4.5.4. We see

that the error increases with dimension to about 5–10% for d = 10. The control

variate improves the results dramatically, the error for d = 10 being now less than

1%.

When using the European control variate we see that the results are biased

upwards, whereas the raw results are biased downwards. This is probably due to

the upward bias introduced by the convexity of the max operator which appears in

the Bermudan and American problems, but not in the European problem.

In one and two dimensions the generator matrix became numerically unstable

for the grid sizes we consider; we have thus not presented results for these low

dimensions here. This lack of convergence is due to the finite precision arithmetic,

and not to instability in the sense that the generator matrix has unstable eigenvalues

(i.e. eigenvalues having positive real part). The method has been found to work

very well in one and two dimensions, but for smaller grid sizes.

4.5.5 Error behaviour

Drawing a parallel with regular grid methods, we expect the error to be related to

δx, the distance between grid points with positive weights in A. In a regular grid

with the same number of points N in each dimension we have n = N d points in

total, and the distance to the nearest point is simply n−1/d. The error when using a

standard finite difference method is of order δx2, or n−2/d.

We thus propose modelling irregular grid errors as in the regular grid case,

allowing for a scalar factor in the exponent as well as a multiplicative factor:

log |ε| = c1 + c2log n

d. (4.5.8)

In Figures 4.5.5 and 4.5.6 we present plots of the log absolute error versus log(n)/d,

and in Tables 4.5.12 and 4.5.13 the regression results. Referring to our assumption

of error behaviour (4.5.8) we find that the complexity is accurately modelled by the

given relationship in all three cases (for suitable c1, c2). The linear relationships

4.5. Experiments 81

observed, on the log scale, strongly suggest that the algorithm has exponential com-

plexity. We note that the behaviour in the Sobol’ and low distortion cases is very

similar, with the European and Bermudan prices showing about the same asymp-

totic relationship, and with American errors showing a slightly faster rate in the

Sobol’ case, although this is barely significant.

The convergence rate for finite difference methods used to solve PDE problems

on regular grids is 1/δx2, or n−2/d which here translates to c2 = −2. From this

point of view our method seems to be slightly slower in convergence than the regu-

lar grid method, although this is barely significant. This may be due to the average

δx being larger as a function of the grid size in the irregular grid case.

The given model for errors implies that the amount of work required to ob-

tain solutions to a certain accuracy increases exponentially with dimension. This

may seem pessimistic in that the curse of dimensionality is not broken; however

the method we use has definite advantages over regular grid methodology in high

dimensions. In particular we note that the number of grid points n can be chosen

freely, the grid points can be adapted to the process density and the number of

boundary points can be substantially reduced for unbounded problems. Regarding

the last point, Section 4.5.7 provides a comparison between the number of bound-

ary points found in regular and normally distributed grids. The results suggest

that the proposed method can handle option pricing problems up to dimension ten,

which sets it aside from traditional finite difference methods which start to become

unwieldy in dimension three or four.

2 3 4 5 6 7 8 9 101

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

d

v

Benchmark 1 × 104 points 1 × 105 points3 × 105 points

2 3 4 5 6 7 8 9 101

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

d

v


Figure 4.5.1: Bermudan pricing results for normal Sobol’ grids presented raw (left)

and using European price as control variate (right).


d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.5370 1.5375 1.5376 1.5376 1.5377 1.5377

4 1.4135 1.4147 1.4155 1.4161 1.4163 1.4166

5 1.3300 1.3329 1.3345 1.3360 1.3365 1.3371

6 1.2532 1.2630 1.2667 1.2757 1.2766 1.2780

7 1.1981 1.2133 1.2137 1.2305 1.2311 1.2313

8 1.1489 1.1664 1.1672 1.1891 1.1938 1.1807

9 1.1116 1.1255 1.1351 1.1530 1.1514 1.1612

10 1.0901 1.1080 1.1078 1.1129 1.1242 1.1218

Table 4.5.2: Results for Bermudan geometric average put options in dimensions

3-10 using normal Sobol’ grids.

d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.5584 1.5588 1.5590 1.5591 1.5592 1.5592

4 1.4332 1.4347 1.4357 1.4362 1.4365 1.4369

5 1.3489 1.3522 1.3537 1.3551 1.3557 1.3563

6 1.2721 1.2818 1.2858 1.2940 1.2951 1.2965

7 1.2182 1.2325 1.2331 1.2482 1.2491 1.2492

8 1.1693 1.1864 1.1870 1.2071 1.2114 1.1993

9 1.1316 1.1460 1.1549 1.1715 1.1700 1.1802

10 1.1102 1.1281 1.1267 1.1324 1.1433 1.1414

Table 4.5.3: Results for American geometric average put options in dimensions

3-10 on normal Sobol’ grids.

4.5. Experiments 83

d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.3461 1.3463 1.3465 1.3465 1.3465 1.3465

4 1.2274 1.2286 1.2293 1.2302 1.2304 1.2304

5 1.1482 1.1505 1.1520 1.1541 1.1545 1.1549

6 1.0716 1.0813 1.0849 1.0977 1.0984 1.0993

7 1.0156 1.0275 1.0318 1.0527 1.0541 1.0545

8 0.9624 0.9792 0.9848 1.0123 1.0151 0.9943

9 0.9231 0.9406 0.9507 0.9735 0.9755 0.9802

10 0.8966 0.9203 0.9277 0.9340 0.9418 0.9424

Table 4.5.4: Results for European geometric average put options in dimensions

3-10 on normal Sobol’ grids.

d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.5382 1.5382 1.5382 1.5379 1.5379 1.5379

4 1.4179 1.4178 1.4180 1.4177 1.4177 1.4179

5 1.3403 1.3409 1.3410 1.3404 1.3405 1.3407

6 1.2892 1.2893 1.2894 1.2857 1.2858 1.2863

7 1.2527 1.2560 1.2521 1.2481 1.2473 1.2470

8 1.2281 1.2288 1.2240 1.2184 1.2203 1.2279

9 1.2074 1.2038 1.2033 1.1984 1.1947 1.1999

10 1.1940 1.1881 1.1805 1.1793 1.1829 1.1799


3-10 on normal Sobol’ grids, using the European price as a control variate.


d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.5595 1.5596 1.5596 1.5594 1.5594 1.5594

4 1.4376 1.4378 1.4382 1.4378 1.4379 1.4382

5 1.3592 1.3602 1.3603 1.3595 1.3597 1.3599

6 1.3082 1.3082 1.3085 1.3041 1.3044 1.3048

7 1.2728 1.2752 1.2716 1.2658 1.2653 1.2649

8 1.2484 1.2487 1.2437 1.2364 1.2379 1.2465

9 1.2274 1.2242 1.2231 1.2169 1.2133 1.2189

10 1.2141 1.2082 1.1994 1.1988 1.2020 1.1994


3-10 on normal Sobol’ grids, using the European price as a control variate.

d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.5372 1.5375 1.5376 1.5377 1.5377 1.5378

4 1.4141 1.4155 1.4160 1.4163 1.4165 1.4166

5 1.3309 1.3338 1.3360 1.3364 1.3370 1.3371

6 1.2695 1.2729 1.2751 1.2777 1.2779 1.2796

7 1.2139 1.2249 1.2255 1.2292 1.2319 1.2321

8 1.1628 1.1773 1.1850 1.1898 1.1899 1.1863

9 1.1234 1.1397 1.1428 1.1548 1.1514 1.1588

10 1.1177 1.1008 1.1131 1.1103 1.1170 1.1242


3-10 using low distortion grids.

4.5. Experiments 85

d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.5583 1.5587 1.5589 1.5590 1.5590 1.5591

4 1.4341 1.4355 1.4361 1.4364 1.4367 1.4369

5 1.3500 1.3528 1.3550 1.3554 1.3561 1.3564

6 1.2875 1.2912 1.2935 1.2961 1.2965 1.2981

7 1.2319 1.2432 1.2433 1.2474 1.2496 1.2502

8 1.1813 1.1952 1.2032 1.2082 1.2080 1.2042

9 1.1412 1.1580 1.1615 1.1730 1.1689 1.1774

10 1.1390 1.1206 1.1315 1.1288 1.1365 1.1434


3-10 on low distortion grids.

d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.3460 1.3463 1.3464 1.3465 1.3465 1.3466

4 1.2287 1.2295 1.2299 1.2301 1.2304 1.2305

5 1.1501 1.1520 1.1535 1.1540 1.1544 1.1546

6 1.0904 1.0947 1.0965 1.0982 1.0987 1.0994

7 1.0394 1.0474 1.0497 1.0523 1.0545 1.0553

8 0.9877 1.0015 1.0078 1.0122 1.0137 1.0131

9 0.9405 0.9605 0.9654 0.9726 0.9729 0.9779

10 0.9080 0.9100 0.9247 0.9291 0.9322 0.9393

Table 4.5.9: Results for European geometric average put options in dimensions

3-10 using low distortion grids.


d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.5379 1.5379 1.5379 1.5379 1.5380 1.5380

4 1.4172 1.4178 1.4179 1.4179 1.4179 1.4179

5 1.3393 1.3403 1.3410 1.3409 1.3410 1.3410

6 1.2867 1.2859 1.2863 1.2871 1.2868 1.2878

7 1.2448 1.2478 1.2461 1.2472 1.2477 1.2471

8 1.2167 1.2174 1.2187 1.2191 1.2178 1.2147

9 1.2017 1.1980 1.1964 1.2011 1.1974 1.1998

10 1.2101 1.1913 1.1888 1.1817 1.1853 1.1853


3-10 using low distortion grids, using the European price as a control variate.

d 5 × 104 10 × 104 15 × 104 20 × 104 25 × 104 30 × 104

3 1.5590 1.5592 1.5592 1.5592 1.5593 1.5593

4 1.4372 1.4378 1.4380 1.4380 1.4381 1.4381

5 1.3584 1.3593 1.3601 1.3599 1.3602 1.3603

6 1.3048 1.3042 1.3047 1.3056 1.3054 1.3064

7 1.2628 1.2661 1.2639 1.2654 1.2654 1.2652

8 1.2352 1.2353 1.2369 1.2376 1.2359 1.2327

9 1.2196 1.2164 1.2150 1.2193 1.2149 1.2184

10 1.2315 1.2111 1.2072 1.2002 1.2048 1.2045


3-10 on low distortion grids, using the European price as a control variate.

Option type c1 c2 R2

European −0.35(±0.23) −1.91(±0.10) 0.971

Bermudan −0.42(±0.14) −1.85(±0.06) 0.988

American −0.55(±0.13) −1.74(±0.05) 0.989

Table 4.5.12: Regression coefficients for the error behaviour on normal Sobol’

grids (95% CI in parentheses).

4.5. Experiments 87

Option type c1 c2 R2

European −0.49(±0.12) −1.94(±0.05) 0.992

Bermudan −0.59(±0.08) −1.84(±0.03) 0.997

American −0.83(±0.08) −1.65(±0.03) 0.995

Table 4.5.13: Regression coefficients for the error behaviour on low distortion grids

(95% CI in parentheses).

2 3 4 5 6 7 8 9 101

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

d

v


2 3 4 5 6 7 8 9 101

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

d

v


Figure 4.5.2: American pricing results for normal Sobol’ grids presented raw (left)


3 4 5 6 7 8 9 10

1.1

1.2

1.3

1.4

1.5

1.6

d

v

Benchmark1 × 105 points3 × 105 points

3 4 5 6 7 8 9 10

1.1

1.2

1.3

1.4

1.5

1.6

d

v


Figure 4.5.3: Bermudan pricing results for low distortion grids presented raw (left)



3 4 5 6 7 8 9 101.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

d

v


3 4 5 6 7 8 9 101.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

dv


Figure 4.5.4: American pricing results for low distortion grids presented raw (left)


1 2 3 4−9

−8

−7

−6

−5

−4

−3

−2

−1

log(n)/d

log(

|ε|)

(a) European

1 2 3 4−9

−8

−7

−6

−5

−4

−3

−2

−1

log(n)/d

log(

|ε|)

(b) Bermudan

1 2 3 4−9

−8

−7

−6

−5

−4

−3

−2

−1

log(n)/d

log(

|ε|)

(c) American

Figure 4.5.5: Log of absolute errors for European, Bermudan and American ge-

ometric average options plotted against log(n)/d for d = 3, . . . , 10 for normal

Sobol’ grids. The points nearly lie in a straight line in all three cases, giving a clear

indication of complexity. See Table 4.5.12 for regression results.

4.5. Experiments 89

4.5.6 Timings

The irregular grid method presented in this chapter can be divided into two compu-

tationally intensive stages: obtaining the generator matrix and performing the time

stepping. The first is the most expensive, but once a matrix has been obtained it

can be reused for a wide range of related problems. We do not consider comput-

ing transition matrices here; it suffices to say that the situation is very similar to

generator matrices.

Here we provide indications of the timings involved; as usual this depends

heavily on the hardware and software used. The software aspect is emphasised

here since there is a huge difference in the performance of different algorithms for

solving the linear programming problem and for solving linear systems of equa-

tions. The experiments are carried out in Matlab on a 866MHz Pentium III under

Windows 2000.

Generator matrix

In dimension d we are interested in solving a large number of linear programming

problems with ηd = d(d + 3)/2 equality constraints and where all variables are

nonnegative. The number of variables needed is not known a priori, but it has

been found that 5ηd is sufficient for points close to the center of the grid, and an

increased number of 20ηd is needed closer to the boundary. The strategy is thus

to order the points according to their norm and try 5ηd neighbours until a certain

failure rate is reached, then to switch to 20ηd neighbours on the remaining points.

In two dimensions a single problem takes about 0.06s and is not sensitive to

the number of variables changing from 5ηd to 20ηd. This is probably due to the

relatively large overhead involved in the Matlab routines. In five dimensions we see

an increase from 0.07s for 5ηd neighbours to 0.10s for 20ηd. In ten dimensions we

see a corresponding increase from 0.31s to 1.90s per problem. It is thus clear that

parallelisation is desirable to keep the computation times reasonable, especially for

higher dimensional problems.

Time stepping

In dimension d and with n grid points we use a generator matrix with n rows each

with ηd + 1 nonzero entries. The complexity of implicit time stepping should thus


be quadratic with dimension and linear with grid size.

For 300,000 points in five dimensions, explicit time steps take about 1.5s and

implicit about 29s with CGS. For ten dimensions, explicit time steps take about

3.0s and implicit about 21s with CGS. The fact that implicit solutions can be faster

in a higher dimension is due to the conditioning of the matrix, making it more

amenable to solution even though it is more dense.

One can thus perform about 10-20 times more explicit than implicit time steps

for the same running time. However there is a tradeoff since the latter generally

give much better precision.

4.5.7 Boundaries

We now compare the observed boundaries presented in Figures 4.5.7 and 4.5.8 to

the naive predictions in Section 4.4.2 and Figure 4.4.2.

The proportion of boundary points goes up quickly with dimension, as pre-

dicted in Section 4.4.2. A simple calculation reveals that the proportion of bound-

ary points for a regular grid with n1/d steps per dimension is 1 − (1 − 2n−1/d)d.

For example, for d = 10 one requires a grid size of about 5 × 1014 to bring the

proportion of boundary points down to 0.5. Using the irregular grid method one

needs about 3 × 105, as seen in Figure 4.5.7.

We cannot compare our results directly to the predictions since we used a max-

imum of 20ηd neighbours when trying to satisfy local consistency. A direct com-

parison would require that we used all points in the grid. It is clear that the observed

boundaries lie at a smaller radius r than the predicted ones. This may be partially

due to the small number of neighbours considered, but may also be caused by the

optimism inherent in the predictions, namely that only the minimum number of

neighbours is required to satisfy the local consistency conditions.

We finally note that the boundaries are not monotone with grid size in Fig-

ures 4.5.7 and 4.5.8. In the case of normal Sobol’ grids, this may be attributed to

the fact that new points in the grid are infeasible with respect to the closest 20ηd

neighbours.

4.6. Conclusions 91

4.6 Conclusions

We proposed a method for pricing options with several underlying assets and an

arbitrary payoff structure. The method was tested for geometric average options,

which can be easily benchmarked, in dimensions three to ten with very accurate

results.

We saw a decay in precision for increasing dimension, a phenomenon which

can be attributed to the increasing distance between points in the approximating

Markov chain, and to the increasing size of the boundary region. An analysis of

the error implies that the method has exponential complexity with dimension, but

the use of control variates was shown to reduce the error substantially. The use

of extrapolation is also expected to provide accurate approximations, although this

was not tested in the present work.

The computation of transition and generator matrices is expensive; however

once generated these matrices can be reused for a large class of similar problems

with time dependent parameters. Furthermore computations are cheap once the

matrix is obtained.

Interestingly we found little difference in complexity between the cases where

Sobol’ and low distortion grids were employed. The complexity observed was

exponential in dimension, of approximately the same order as regular grid discreti-

sations.

Although the method extends naturally in principle to arbitrary Markov pro-

cesses with parameters depending on state and time, further extensions to the nu-

merical procedures are required to make the proposed method computationally at-

tractive in such cases. For example, this is of interest when considering Bermudan

swaptions where the drift is state dependent.


1 2 3 4−9

−8

−7

−6

−5

−4

−3

−2

−1

log(n)/d

log(

|ε|)

(a) European

1 2 3 4−9

−8

−7

−6

−5

−4

−3

−2

−1

log(n)/d

log(

|ε|)

(b) Bermudan

1 2 3 4−9

−8

−7

−6

−5

−4

−3

−2

−1

log(n)/d

log(

|ε|)

(c) American

Figure 4.5.6: Log of absolute errors for European, Bermudan and American geo-

metric average options plotted against log(n)/d for d = 3, . . . , 10 for low distor-

tion grids. The points nearly lie in a straight line in all three cases, giving a clear

indication of complexity. See Table 4.5.13 for regression results.

0.5 1 1.5 2 2.5 3

x 105

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

n

r

d=3

d=5

d=10

(a) Radius

0.5 1 1.5 2 2.5 3

x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

n

p

d=3

d=5

d=10

(b) Proportion

Figure 4.5.7: Smallest norms of points in normal Sobol’ grids for which local

consistency could not be satisfied and proportion of points in the boundary region

with 20ηd nearest neighbours. Compare Figure 4.4.2.

4.6. Conclusions 93

0.5 1 1.5 2 2.5 3

x 105

2.5

3

3.5

4

n

r

d=3

d=5

d=10

(a) Radius

0.5 1 1.5 2 2.5 3

x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

n

p

d=3

d=5

d=10

(b) Proportion

Figure 4.5.8: Smallest norms of points in normal low distortion grids for which

local consistency could not be satisfied and proportion of points in the boundary

region with 20ηd nearest neighbours. Compare Figure 4.4.2.

Chapter 5

A Method Using Interpolation

5.1 Introduction

The prevalence of high-dimensional Bermudan-style derivative contracts in world

markets, particularly those based on interest rates, has exposed a need for effi-

cient algorithms which provide accurate price estimates. The Bermudan interest

rate swap option (swaption) is an important practical example of such a high-

dimensional problem.

While much focus in recent literature has been placed on dimensionality re-

duction, it has been argued for example in Longstaff et al. [49] that the application

of single factor exercise strategies may result in significant losses. The necessity

for multifactor models is also emphasised in Sidenius [67], who finds a strong sen-

sitivity of derivative prices to the number of factors used in the model. He further

argues that a large number of factors is required to obtain stationary estimates of the

volatility term structure, which is what one typically expects to see in the market.

We therefore make the underlying assumption that we are dealing with a prob-

lem whose dimensionality cannot be reduced to a level which is manageable thr-

ough the use of traditional methods. In practise this means we focus on Bermudan

problems depending on at least three or four factors.

The value of such contracts, given the risk-neutral dynamics and payoff struc-

ture, is easily formulated as a dynamic programming problem using a continuous

state space and a number of time steps corresponding to the number of exercise op-

portunities. In solving such problems on a digital computer, however, one faces the

95

96 Chapter 5. Method Using Interpolation

challenge of approximating the continuous problem by a discrete one whose solu-

tion is close to the true solution and which can be solved in a reasonable amount of

time.

There are several ways of performing such a reduction, but only a few are com-

putationally attractive for high-dimensional problems. Carriere [20] and Longstaff

and Schwartz [50] propose and test methods based on first estimating the optimal

stopping rule. Once a stopping rule is found, the problem of finding the value

becomes one of integration rather than optimisation, and an unbiased estimate of

a lower bound on the option value (given that the approximate stopping rule is

generally suboptimal) may be obtained through Monte Carlo simulation.

A dual approach, of which variants are proposed by Rogers [64], Haugh and

Kogan [37] and developed further by Jamshidian [44] and Kolodko and Schoen-

makers [45], approaches the problem from the hedging perspective. In this method

one searches for an optimal martingale which, when viewed as the relative value of

a hedging portfolio, minimises the cost of replicating the claim. Again, once such

a martingale is found, the problem becomes linear, and now an unbiased estimate

of an upper bound on the option value (given that the approximate hedging strategy

is also generally suboptimal) may be obtained through simulation.

These methods are attractive in that they provide estimates with known biases,

but both methods require optimisation over a functional space; the stopping rule

being a function of the many underlying factors, and the hedging portfolio being

a martingale based on these factors. Finding approximations to the optimal strate-

gies usually involves searching in a finite-dimensional class parameterised by some

basis, which must be chosen in a clever manner.

Our objective here is to estimate prices of Bermudan swaptions in the LIBOR

market model (LMM). The LMM was developed by Jamshidian [43] and Brace

et al. [15], and models the forward LIBOR rates which may be observed directly

in the market. This model has been favoured over its sibling, the swap market

model (SMM), because of the relative ease of pricing swap contracts in the LMM

as opposed to pricing caps in the SMM. This is explained in some detail in Pietersz

and Pelsser [63]. Geometric average options are also investigated, since accurate

benchmarks are available through reduction to a one-dimensional problem.

Numerical methods for swaption pricing in the LMM have been treated by

several authors, including Andersen [1] and Pedersen [61]. The predominant ap-

5.1. Introduction 97

proach is in line with that of [20, 50], namely to first find an exercise rule and then

to estimate the swaption value through simulation. We use the results of [1] as

benchmarks in our experiments.

We propose an intuitive method based on a state space discretisation of the

dynamic programming problem. Tsitsiklis and Van Roy [71] suggest such a me-

thod for high-dimensional Bermudan problems where the continuation values are

determined using dynamic programming together with projections onto a space

of features. Our approach differs however in that we do not specify features, but

consider separate interpolation and approximate quadrature operators acting on the

value function at each time step.

The approximate quadrature operator is constructed using standardised quasi-

Monte Carlo (QMC) draws, as suggested in [21], for the geometric average options.

The use of low discrepancy methods becomes more difficult in the LMM due to

the time- and state-dependency of the parameters; in this case we propose antithetic

simulations.

The interpolation operator seems to be a more difficult problem. We investi-

gate two methods: nearest neighbour interpolation and local quadratic interpolation

(linear interpolation was also investigated, but the results are not presented as the

method produced very large biases).

Nearest neighbour interpolation is perhaps the simplest method, producing

piecewise constant interpolants whose values may be determined quickly through

the use of fast nearest neighbour searching techniques.

Local polynomial interpolation, or moving least squares, has been a topic of

active research recently in the literature on mesh-free methods. Levin [47] and

Wendland [72] show that such interpolants have a precision proportional to hm

where the fill distance h is a measure of the grid resolution and m is the order

of the polynomials used. Unfortunately the method is very slow when applied

directly due to the need for matrix inversion at each interpolation point. Maz’ya and

Schmidt [53] propose an interesting matrix-free method for irregular grids which

avoids the computational complexity, and much inspiration may be drawn from the

literature on nonparametric statistics. We do not investigate such extensions here

however.

Fasshauer et al. [30] and Hon [39] have applied moving least-squares methods

to option pricing problems with some success; they do not however consider high-


dimensional problems.

We proceed to introduce the LMM and swaption pricing problem in Section

5.2. We present our methodology in Section 5.3 and experiments in Section 5.4.

Conclusions are drawn in Section 5.5.

5.2 Bermudan swaption pricing in the LIBORmarket model

In the LMM, forward rates are used to set up an arbitrage-free and easily calibrated

formulation of the term structure. The pricing of swaption contracts in the LMM

is by no means straightforward, especially when the contract allows early exercise.

5.2.1 Setup of the LMM

We introduce here the basic concepts of the LMM, referring the reader to Brace

et al. [15] and Jamshidian [43] for the original presentation, and to Pelsser [62],

Andersen and Andreasen [2] and Brigo and Mercurio [17] for more recent devel-

opments.

Consider the time steps t0 < · · · < tK (these are reset dates for the tenor

structure), where each tk may be thought of as referring to a swaption exercise

opportunity. Let αk = tk+1 − tk be the day-count fraction between reset dates and

n(t) the next reset date function defined such that tn(t)−1 ≤ t < tn(t).

Denote by Dk(t) the price at time t of a discount bond maturing at tk; the

LIBOR rate corresponding to the period (tk, tk+1) is then

Lk(t) =1

αk

Dk(t) −Dk+1(t)

Dk+1(t). (5.2.1)

One may also express the Dk in terms of the Lk :

Dk(tj) =

k−1∏

i=j

(1 + αiLi(tj))−1. (5.2.2)

In the LMM, we assume that each Lk(t) follows a diffusion law. All active

rates Lk, k = n(t), . . . ,K , are brought under the terminal measure (the measure

5.2. Bermudan swaption pricing 99

under which LK is a martingale) through a change of numeraire and application of

Girsanov’s theorem. This leads us to consider the correlated diffusion

dLk(t)

Lk(t)= µk(L(t), t)dt + σk(L(t), t)dW (t) (5.2.3)

where L(t) = (L0(t), . . . , LK(t))′, W (t) is a Brownian motion under the terminal

measure having dimension d ≤ K and σk are d-vectors giving the volatilities and

correlations. The risk-neutral drift for each Lk is given by

µk(L(t), t) = −K∑

i=n(t)

αiσi(L(t), t)Li(t)

1 + αiLi(t)σk(L(t), t), (5.2.4)

and the σi are chosen appropriately, for example to fit market data within a param-

eterised setting.

5.2.2 Swaption pricing

An interest rate swap is a contract allowing the holder to exchange one set of in-

terest rate payments for another. In the present work we limit the presentation to

payer swaps, where one exchanges a floating set of payments for fixed payments.

A European swaption is then a contract which gives the holder the right, but not the

obligation, to enter into a swap contract at a certain future date, and a Bermudan

swaption allows the holder to enter into a swap contract at any one of a prespecified

set of dates (though not more than one).

We further limit our discussion to the case where the floating payments are

determined by the LIBOR forward rate process. The value of a swap contract is

determined by the discount bonds related to payment dates of the swap at time tkas

vk(tk) = 1 −DK(tk) −KK−1∑

i=k

αiDi+1(tk) (5.2.5)

where tK is the final maturity of the swap contract. Since bond prices may be de-

termined from LIBOR rates through (5.2.2), the swap value may also be expressed

through the LIBOR rates Lk(t).

A European swaption expiring at tk takes the value ψk(tk) = max(vk(tk), 0)

at time tk, and at times t < tk

vk(t) = DK(t)EK

(

ψk(tk)

DK(tk)

)

(5.2.6)


where the expectation is taken under the terminal measure. We thus see that in the

European case the swaption value is an expectation of a known payoff with respect

to an unknown density.

In the Bermudan case we add an extra complication to the problem in the form

of multiple exercise opportunities. As in Andersen [1], we characterise the Bermu-

dan contract with three dates: the first date where exercise is allowed, ts (otherwise

known as the lockout date); the last date where exercise is allowed, tx; and the final

swap maturity te. We assume that t0 < ts < tx < te, and in the examples we only

consider cases where x = e− 1. We now formulate the value of such a Bermudan

swaption as

vk(t) = DK(t) supτ∈Ts,x

EK

(

ψk(τ)

DK(τ)

)

(5.2.7)

where Ts,x is the set of stopping times on ts, . . . , tx with respect to the natural

filtration. The Bermudan swaption value is thus the expectation of a known payoff

with respect to an unknown density, where the expectation is evaluated using the

optimal stopping rule τ .

Neither the European nor the Bermudan case admits a closed form solution,

and thus numerical methods must be employed to find approximate values.

5.3 Methodology

We focus on Bermudan options with a fixed set of exercise opportunities. Values

are computed only at the exercise dates, the continuation value being approximated

using numerical integration of the value function at the following exercise oppor-

tunity, which is extended to the entire state space using interpolation.

We first present a solution method which is computationally tractable for the

kind of problem presented in Section 5.2. We prove convergence of this method

under certain conditions.

5.3.1 Framework and assumptions

In order to simplify the analysis, we assume that any discounting is incorporated

in the value functions, so that all value functions are given in time zero currency

units.

5.3. Methodology 101

We assume exercise is possible at times tk for k = 1, . . . ,K and where

0 = t0 < t1 < · · · < tK = T. (5.3.1)

The case where exercise is not possible at t = 0 can be taken into account with

obvious modifications in the following.

We consider problems where, at each exercise opportunity tk, the value func-

tion can be represented as vk : Ωk → R where Ωk ⊆ Rdk for some appropriate

dimension dk.

At each exercise opportunity we allow an arbitrary grid

Xk = xk,1, . . . , xk,nk ⊂ Ωk (5.3.2)

on which the value function is to be estimated, where the number of points in

grid Xk is nk. Our approximation to the value function at time tk and state xk,i

is denoted vk,i. Clearly the grids Xk should be chosen with importance sampling

considerations in mind.

The most difficult part of implementing a dynamic programming algorithm for

Bermudan options computationally is in estimating the continuation values. We

now introduce the two operators which will be used to estimate these.

Let Qtk ,x be the appropriate quadrature operator for time tk and state x given

by

Qtk ,xf =

∫

Ωk+1

f(y) dp(X(tk+1) = y|X(tk) = x) (5.3.3)

whereX(t) is the stochastic process followed by the underlying variables and p(·|·)

is the conditional density implied by the process. Now define Qtk ,x,m to be a

numerical approximation to Qtk,x where m is a parameter affecting the precision

of Qtk ,x,m. For notational compactness we also define the entire quadrature and its

approximate version, respectively, as

Qk(x) = Qtk ,x (5.3.4)

Qk,m(x) = Qtk ,x,m. (5.3.5)

Further, let IX be an interpolation operator taking a vector of function values

f defined on the grid X , and returning a function in some class CI , and defined on

the entire state space,

IX : RX → CI(Ωk,R). (5.3.6)


For this purpose we define vk as the vector having ith entry vk,i. Continuation

value estimates are denoted ck,i, and ck is the vector with ith entry ck,i.

We also define the linear operator PX to be the projection operator taking a

function and returning a vector of values of the function at the grid points in X :

PX f = (f(x1), . . . , f(xnk))′ . (5.3.7)

For ease of notation we denote Pk = PXkand Ik = IXk

.

We note that the true value functions at the exercise opportunities are given by

Algorithm 5.3.1. We propose solving the dynamic programming problem compu-

tationally as shown in Algorithm 5.3.2.

Algorithm 5.3.1 Exact algorithm for finding Bermudan option price where vk(·)

are the value functions, ck(·) are the continuation values, ψk(·) are the payoff func-

tions and Qk are the entire quadrature operators, all for each time tk.vK(·) := ψK(·)

for k = K − 1, . . . , 0 dock(·) := Qk vk+1(·)

vk(·) := max(ψk(·), ck(·))

end for

Algorithm 5.3.2 Numerical algorithm for finding Bermudan option price, where

the vk are the vectors of approximate values, ck are the estimated continuation

values, ψk are the vectors of (exact) payoffs and Qk,m are the approximate entire

quadrature operators, all for each time tk.

vK := ψK

for k = K − 1, . . . , 0 dock := PkQk,m Ik+1 vk+1

vk := max(ψk, ck)

end for

Note that, since the value function at expiry vK is known exactly, the interpo-

lation operator in Algorithm 5.3.2 need not be applied at step k = K − 1; instead

IX vK is replaced by vK . This implies further that the grid XK need not be used in

practise.


5.3.2 Convergence

The following elementary lemma states that a maximum operator of the sort ap-

plied in Algorithm 5.3.2 does not increase the accumulated error.

Lemma 5.3.1 Let b, c and c be real numbers. Define

a = max(b, c)

a = max(b, c).

Then

|a− a| ≤ |c− c|. (5.3.8)

Remark 5.3.1 Lemma 5.3.1 merely states that the max function is Lipschitz in

both of its arguments.

Before stating the convergence theorem, we state some conditions that will be

used. We assume that the operators Qk,m and Ik are consistent in that, for every

εQ,m, εI > 0 we can find m, nk such that for all m > m, nk > nk

|Qk,mvk+1 −Qkvk+1|∞ ≤ εQ,m (5.3.9)

|IkPkvk − vk|∞ ≤ εI (5.3.10)

for all k where vk is defined in Algorithm 5.3.1. We also assume the operators are

Lipschitz continuous in that there exist constants cQ,m, cI such that

|Qk,mf |∞ ≤ cQ,m|f |∞ (5.3.11)

|Ikg|∞ ≤ cI |g|∞ (5.3.12)

for all k where f ∈ CI + linspanvk+1 and g is any nk-vector. We also assume

Qk,m to be linear in that

Qk,m(f + g) = Qk,mf + Qk,mg (5.3.13)

for all k, m.

Theorem 5.3.1 Under the conditions (5.3.9)–(5.3.12), the approximations vt0 ,xi

defined through Algorithm 5.3.2 converge uniformly to the exact solutions v(t0, xi)

of Algorithm 5.3.1 as m, nk → ∞.


ing (5.3.17) recursively, we obtain

|v0 − v0|∞ ≤(

1 + cQ,mcI + · · · + (cQ,mcI)K−1)

(εQ,m + cQ,mεI) .

(5.3.18)

Hence choosing the m and nk large enough allows us to achieve an arbitrarily

small error.

Note that in the preceding theorem, one may reduce the error estimates by

choosing operators which have small continuity constants cQ,m, cI .

The consistency condition on the interpolation operator (5.3.10) is very strong.

In particular, due to our use of the L∞ norm, (5.3.10) stipulates a uniform bound

on the interpolation error over the entire state space. On the contrary, interpolation

errors for areas of the state space which fall in the tails of the (unconditional)

density should only have a minor effect on the accuracy of the solution, and thus

greater errors may be tolerated in these tail areas. We now propose a framework

which takes this into account by using measures which are adapted to the process

density.

Consider now a sequence of measures µk relating to tk for k = 0, . . . ,K ,

which will be used in estimating the accumulated error. We define discrete coun-

terparts of these measures µk which are related to the grids Xk in that there exists

a constant cP such that

|Pkf |µk≤ cP |f |µk

(5.3.19)

for all k where PX is the linear projection operator given in (5.3.7).

The approximate quadrature and interpolation operators are now assumed to

be consistent in that for any εQ,m, εI > 0 we can find m, nk such that for all

m > m, nk > nk

|Qk,mvk+1 −Qkvk+1|µk≤ εQ,m (5.3.20)

∣

∣IkPkvk − vk

∣

∣

µk≤ εI (5.3.21)

for all k. We further assume the existence of continuity constants cQ,m, cI such

that

|Qk,mfk+1|µk≤ cQ,m|fk+1|µk+1

(5.3.22)

|Ikg|µk≤ cI |g|µk

(5.3.23)


for all k and where fk+1 ∈ CI +linspanvk+1 for each k, and g is any nk-vector.

As in (5.3.13) above, Qk,m is again assumed to be linear.

Theorem 5.3.2 Under the conditions (5.3.13), (5.3.19)–(5.3.23), the approxima-

tions vt0,xidefined through Algorithm 5.3.2 converge in the sense of µk to the exact

solutions v(t0, xi) of Algorithm 5.3.1 as m, nk → ∞.

Proof. The proof is similar to that of Theorem 5.3.1, except for the different

norms and that the inequality in the first line of (5.3.17) is no longer valid in gen-

eral. We now have

|Qk,mIk+1vk+1 −Qkvk+1|µk≤ |Qk,mvk+1 −Qkvk+1|µk

+ |Qk,mIk+1Pk+1vk+1 −Qk,mvk+1|µk

+ |Qk,mIk+1vk+1 −Qk,mIk+1Pk+1vk+1|µk.

(5.3.24)

Using Lemma 5.3.1, conditions (5.3.13), (5.3.19)–(5.3.23) and (5.3.24) we

have

|vk −Pkvk|µk≤ |Pk (Qk,mIk+1vk+1 −Qkvk+1)|µk

≤ cP |Qk,mIk+1vk+1 −Qkvk+1|µk

≤ cP

(

εQ,m + cQ,mεI + cQ,mcI |vk+1 −Pk+1vk+1|µk+1

)

.

(5.3.25)

Applying (5.3.25) recursively and noting vK −PKvK ≡ 0 gives

|v0 − v0|µ0≤ cP

(

1 + cQ,mcI + · · · + (cQ,mcI)K−1)

(εQ,m + cQ,mεI) .

(5.3.26)

Remark 5.3.2 The consistency conditions (5.3.9), (5.3.10), (5.3.20) and (5.3.21)

may be difficult to verify in practise. In fact these are joint conditions on the

operators and the function class to which the solutions vk belong; the conditions

being easier to satisfy for more regular functions. As mentioned above, conditions


(5.3.9) and (5.3.10) are rather strict and are not expected to hold for many practical

examples. Conditions (5.3.20) and (5.3.21) on the other hand may be expected

to hold for a much wider class of functions, for given operators, and where the

measures µk are chosen appropriately.

5.3.3 Approximating the quadrature operator

The quadrature operator as defined in (5.3.3) is an integral with respect to the con-

ditional density of the process, which is in most cases not known explicitly. The

most general case which yields tractability is where, given that X(tk) = x, we

may simulate values of X(tk+1). We base our construction of the quadrature ap-

proximation Qtk ,m on this case.

Note however that the LMM does not satisfy this assumption strictly, although

one may obtain simulated values which are distributed arbitrarily well by decreas-

ing the time step.

We consider the following quadrature rule

Qtk,x,mf =1

m

m∑

i=1

f(Xi) (5.3.27)

where the i.i.d. variables Xi ∼ p(X(tk+1) = y|X(tk) = x) may be evaluated

through simulation. This quadrature approximation does not satisfy the assump-

tions of consistency (5.3.9) strictly, but for some εQ,m, m may be chosen such that

the condition is satisfied with a probability close to 1. The continuity (5.3.22) and

linearity (5.3.13) assumptions are clearly satisfied.

The literature on QMC methods suggests that in many cases the finite-sample

precision of this quadrature operator may be dramatically improved by using low

discrepancy numbers in the simulation procedure. The use of QMC or low distor-

tion points [3] may also lead to deterministic error bounds in (5.3.27). If neither

QMC nor low distortion methods are applicable, one is usually able to incorporate

variance reduction techniques such as antithetic variables in simulations.

5.3.4 The interpolation operator IX

The interpolation operator represents the manner in which we extend information

from the grid to a function on the entire state space. Clearly such a function must


be well-behaved in relation to the value function, as expressed in the consistency

and continuity conditions of Section 5.3.1.

The consistency condition (5.3.10) for Theorem 5.3.1 is perhaps the more dif-

ficult of these assumptions to satisfy, in particular due to the unbounded domain.

The respective condition (5.3.21) for Theorem 5.3.2 is easier to satisfy. Consider

for example the nearest neighbour interpolation[

IX f]

(x) = fargmini|x−xi| (5.3.28)

which only satisfies (5.3.10) when the function to be approximated converges to

a constant as |x| → ∞, but which satisfies (5.3.21) so long as the function to be

approximated is integrable with respect to µk, and the Xk are chosen appropriately.

Another candidate, which has received much attention recently in the litera-

ture on mesh free methods, is the local polynomial reproduction, or moving least-

squares method. This is a class of methods using information from nearby points to

form an interpolant, and has previously received attention in the option pricing lit-

erature in Hon [39] and Fasshauer et al. [30]. These papers do not however address

the problem of high-dimensionality.

The method has received theoretical support in the case of irregularly-spaced

points in the work of Levin [47] and Wendland [72]. To implement the local poly-

nomial reproduction method on the grid X = x1, . . . , xn, one first chooses a

class of polynomials; for example we may choose the class Rdm of polynomials

in d variables having degree less than or equal to m. We then look for weight

functions wj such that∑

j

p(xj)wj(x) = p(x) (5.3.29)

for all p ∈ Rdm. One then forms the (quasi-)interpolant

[

IX f]

(x) =∑

j

wj(x)f(xj). (5.3.30)

The extension operator in (5.3.30) is in general a quasi-interpolant because it does

not necessarily satisfy the value reproduction condition [IX f ](xj) = fj . The

weight functions wj(x) are assumed to be local by construction in that, for x far

away from xj , we have wj(x) = 0.

We implement the local polynomial reproduction method for the class of quad-

ratic functions Rd2 as follows. First, assume that ξd = dimRd

2 n, that is the


number of grid points is much greater than the number of points required to fit

quadratic functions. Now, for each point x the weight functions at that point are

determined as follows: choose a number of nearest neighbours η ≥ ξd to use for

fitting the local quadratic; only these η neighbours will receive nonzero weights in

(5.3.29). The weights are calculated for x ∈ Rd through solving the underdeter-

mined system

p0(x1) · · · p0(xη)...

...

pξd(x1) · · · pξd

(xη)

w1(x)...

wη(x)

=

p0(x)...

pξd(x)

(5.3.31)

for the wj(x) where p0, . . . , pξdis a basis for Rd

2. In our experiments, we use the

basis p0 = 1, p1 = x1, . . . , pξd= x2

d, and the nonzero wj(x) are determined by

the columns of the matrix on the LHS having the largest norms. In this case we can

thus conclude that the weight functions wj(x) are piecewise polynomial, where

the pieces are exactly those regions of the state space having the same nearest

neighbour set.

We note that this method is computationally intensive for moderate dimen-

sions. In this respect promising developments are currently being made in the area

of matrix-free methods for local polynomial interpolations, leading to much faster

computing times (see for example [53, 29]). These methods have not been in-

vestigated for high dimensional problems however, so their effect on the current

analysis is an open issue.

5.3.5 Variance reduction

There are a variety of variance reduction techniques which may be employed when

using Algorithm 5.3.2. We comment on the techniques that will be employed for

the experiments in Section 5.4.

For Bermudan option problems, there is almost always an effective control

variate available in the form of a European counterpart, whose value can usually be

found through simulation. To be effective, a control variate should have a high cor-

relation with the problem at hand. For American options on stocks, the European

option expiring at the same time as the American is often effective; for Bermudan

swaptions, the European swaption expiring at the next reset date is often effective.


We consider here only inner control variates, as opposed to the more standard

outer control variates. This means that the control variate is applied at each time

step, and not only once at the end of the experiment.

For the quadrature operator Qtk ,x,m, a standard technique is to sample using a

set of low discrepancy points. Asymptotically this leads to a convergence rate of

close to n−1 rather than the usual n−1/2 for standard Monte Carlo sampling. We

have found for small samples that one can often improve the quadrature estimates

markedly by normalising the first and second moments of the low discrepancy

points, where this is possible. This technique is also suggested in Carriere [21],

where it is found to improve finite sample behaviour in pricing a Bermudan put

option.

For example, suppose we have generated the low discrepancy point set X0 =

x0,1, . . . , x0,m, to be used in the quadrature. We first normalise X0 with respect

to the first moment; let X1 = x1,1, . . . , x1,m where

x1,i = x0,i −1

m

∑

j

x0,j . (5.3.32)

The point set X1 now has mean zero. Let us now normalise X1 with respect to the

second moment; let X2 = x2,1, . . . , x2,m where

x2,i = R−1X1x1,i (5.3.33)

where RX1is the Cholesky factor of ΣX1

= 1m

∑

j x1,jx′1,j . The point set X2 now

has zero first moment and unit second moment.

This method of normalisation is related to the ideas of local consistency pre-

sented in [46] and Chapter 4 (also published as [10]), in which transition probabil-

ities to a given set of points are determined such that the first and second moments

are matched. Here we first set the probabilities (namely equal weights adding to

unity) and then select the points to be used in a locally consistent manner. Note

that asymptotically the behaviour of this method is uncertain; we advocate its use

only for small samples.

When standard low discrepancy point methods become less effective (e.g. for

path dependent processes), we use antithetic variables.

As another possible method to improve accuracy in Bermudan problems, we

consider breaking down the value function into the sum of the intrinsic value and

5.4. Experiments 111

the early exercise premium. The former may be integrated directly, whereas the

latter must first be interpolated.

5.3.6 Multiple grids

In order to further smooth the interpolation, one may consider using multiple grid

solutions at each time step. This is expected to improve solution behaviour, espe-

cially in the case of nearest neighbour interpolation where the interpolant is most

nonsmooth.

Briefly, one chooses a number of (different, possibly randomised) grids ` > 0

to be used. At each time step, one forms approximations on each grid by applying

the quadrature operator to the average of the interpolants at the previous time steps.

The value function approximation is thus implicitly represented as an average of

the ` grid solutions.

The reader is left to make the necessary adjustments to Algorithm 5.3.2.

5.3.7 Parallelism

It should be emphasised that all the methods suggested here are embarrassingly

parallel in nature; that is, one can obtain an almost linear speedup by employing

multiple processors. This is a result of the fact that the suggested operators can be

evaluated in parallel when estimating the value function at separate points xi.

5.4 Experiments

Numerical experiments are conducted for two examples. First we look at geo-

metric average options in a Black-Scholes market with 1-10 dimensions. This is a

useful test since, although geometric average options do not constitute an important

application, accurate benchmarks are readily available for problems with arbitrary

dimension. We refer the reader to Chapter 4 for derivations in the following setting.

Second, we look at Bermudan swaptions in a LIBOR setting. This is currently

one of the most challenging applications in mathematical finance because of its

high dimensionality and because of the time and state dependence of the param-

eters in the risk-neutral process. It is also an important application due to the

widespread use of such contracts.


5.4.1 Bermudan geometric average options

The setting considered is the same as that in Chapter 4, to which the reader is

referred for the benchmark calculations and discussion. We only present graphical

results for this setting in the current work.

Briefly, we consider assets with values Si(t) = eX(t) evolving according to

a d-dimensional correlated geometric Brownian motion with corresponding risk

neutral process

dX(t) =

(

r11 −1

2diag Σ

)

dt+RdW (t) (5.4.1)

for risk-free rate r and R′R being the Cholesky decomposition of the covariance

matrix Σ = (σiσjρij). We assume a time horizon T and payoff function

ψ(S) =

(

K −(

∏

Si

)1/d)+

(5.4.2)

where K is the strike price.

We assume starting asset values Si(0) = 40 for all i and strike price 40. The

risk-free rate is taken as 0.06, the volatilities σi = 0.2 for all i, and correlations

ρij = 0.25, i 6= j. Figures 5.4.1-5.4.4 show how the numerical method performs

in predicting the early exercise premium, with increasing dimension.

We use a standardised, randomised low discrepancy quadrature operator, as

discussed in Section 5.3.5. Ten trials are performed for each dimension and number

of grid points, the results differing due to the randomisation involved in each case.

We investigate the mean and range of the result obtained.

In Figure 5.4.1, we see the relative performance of using local quadratic and

nearest neighbour interpolations; the randomisation in these cases being restricted

to the quadrature operator and calculation of the control variate. For all cases, the

deterioration of precision with dimension is clear. With local quadratic interpola-

tion one sees reasonable results with 500 points, whether a control variate is used

or not. With nearest neighbour interpolation we see a high bias for low dimen-

sions and a low bias for higher dimensions, suggesting overall a combination of

biases. Once the control variate is applied the results show a low bias for most

cases, particularly in higher dimensions.

Figure 5.4.2 presents timings results, showing clearly the difference in com-

putational complexity between the two interpolation methods. Indeed the nearest

5.4. Experiments 113

neighbour method shows linear complexity, whereas the local quadratic method

shows a much higher complexity. For 500 grid points in dimension ten, it takes

nearly 25 minutes per experiment; for this reason we did not investigate local

quadratic interpolation with 1000 points, and we restrict further experiments to the

nearest neighbour method. We reserve the faster approximate methods of [53, 29]

for future investigation.

Figure 5.4.3 shows results using the nearest neighbour method where the grids

are randomised, and where multiple grids are used. These randomised grid meth-

ods show a high bias for higher dimensions, but when the control variate is applied,

the bias is much reduced and the average result is accurate to within one cent for

both 500 and 1000 point grids. The results become tighter when three grids are

used as opposed to a single grid, although the average result is not affected greatly.

Figure 5.4.4 shows the effect of treating the early exercise premium and intrin-

sic value separately. Since interpolation is not required to integrate the intrinsic

value, one may expect a performance increase in this case. In fact we see that the

bias is reversed in this case, and the results become slightly tighter for the ran-

domised cases; the improvement is not great however. The reason for the bias

reversal can be explained in terms of the bias introduced through interpolation; in

the original experiments the interpolated function is everywhere convex, whereas

in Figure 5.4.4 the function has a downward kink at the exercise boundary.

5.4.2 Bermudan swaptions

For our experiments on the pricing of Bermudan payer swaptions, we follow the

examples of Andersen [1]. Using the notation of [1], we recall the two volatility

scenarios with one and two factors, respectively:

Scenario C:

σk(t) = 0.2 ∀k, t ≤ tk; (5.4.3)

Scenario D:

σk(t) =(

0.15, 0.15 −√

0.009(tk − t))′

∀k, t ≤ tk. (5.4.4)

We consider four of the contracts presented in [1]; the details are given in Table

5.4.1. The time between reset dates in the swaption is always α = 0.25, and the

starting LIBOR rates are assumed flat at Lk(0) = 10% for all k = 0, . . . , te − 1.


Contract ts tx te

I 0.25 1.25 1.50

II 1.00 2.75 3.00

III 1.00 5.75 6.00

IV 3.00 5.75 6.00

Table 5.4.1: Lockout time (ts), final exercise time (tx) and final swap maturity (te)

of the Bermudan swap option contracts.

As in [1], we use four time steps between reset dates and an Euler discretisation

in the simulation of the LIBOR process. The dimensionality of the contracts thus

ranges from 5 for Contract I to 23 for Contracts III and IV.

For the control variate, the European swaption expiring at the next reset date

was found to perform well.

We use 100 antithetic points and antithetic paths simulations in the quadrature.

Antithetic paths were used due to the standardised QMC points losing their ef-

fectiveness with the time- and state-dependent parameters; knowledge of the first

and second moments of the quadrature density could have helped in this respect.

Further, 500 antithetic points were used in the inner control variate calculations.

To determine benchmarks for the early exercise premium in the experiments,

we use the difference between the Monte Carlo results found in Andersen [1] for

the European and Bermudan prices. The results for exercise strategy (I), as defined

in [1], are used for the Bermudan.

Figures 5.4.5 and 5.4.6 show numerical results for Bermudan swaption pricing.

Ten experiments were conducted for each contract, each grid size and each strike.

One sees that results from a single experiment may not always be accurate, how-

ever on average the results are accurate for contracts I-II and slightly high-biased

for contracts III-IV. The latter is in agreement with the geometric average pricing

results, which found a high bias when integrating the value function directly. This

also confirms that the longer-maturity contracts, which involve more state variables

and more steps in the dynamic programming algorithm, are more difficult to price

accurately.

The figures show that varying the number of grid points in the range 100-500

does not affect the results greatly, which seems surprising given that the amount of

work varies five-fold over this range.

5.5. Conclusions 115

The timings presented in Figure 5.4.7 show that the work depends linearly on

the number of grid points, as would be expected. We also see that the contracts

with more exercise opportunities, in particular contract III, are computationally

more intensive.

5.5 Conclusions

We have proposed a class of dynamic programming algorithms for solving Bermu-

dan option pricing problems. The dynamic programming problem uses an irregular

grid at each time step on which the value function is approximated. Continuation

values are approximated by applying an interpolation and a quadrature operator.

Convergence of the algorithm has been demonstrated provided that the operators

satisfy certain boundedness, continuity and linearity conditions.

In the experiments, we find that using local quadratic approximations is very

intensive computationally. This computational burden may possibly be alleviated

through the use of “approximate approximations” as advocated by Maz’ya and

Schmidt [53]; indeed these methods should improve the computational complexity

significantly, but at an unknown cost in terms of accuracy.

In terms of complexity, it was noted that the algorithm presented is easily par-

allelisable; thus the computational time is limited by the number of processors

available. Thus solutions may be computed much faster than what is indicated by

the timings given in this chapter.

We find that using an inner control variate can improve results markedly, and

indeed seems necessary to obtain reasonably unbiased estimates in our tests. We

find that applying the interpolation operator directly to the value function leads

to high-biased estimates; on the other hand, separating the value function into its

intrinsic and early exercise components and applying the interpolation operator to

the early exercise premium leads to low biased estimates. These biases are not

surprising, since in the first case we are interpolating a convex function and in the

second case a concave function.

What may be surprising is the relatively low number of grid points required

to obtain an accurate solution. For example, the geometric average Bermudan put

options, 500 points seemed sufficient with the use of an inner control variate; for

the Bermudan swaptions there was little difference between the solutions from


using 100 and 500 points. In the latter case it is not possible to make a strong

statement about accuracy due to the absence of benchmarks. In the former case the

accuracy depends on the use of an inner control variate, as shown in the relevant

figures.

A comparison with the results of Andersen [1] and Pedersen [61] shows agree-

ment of the price estimates to within a small margin of error. Since no benchmarks

are available, we cannot give a definitive report on the accuracy; we do know how-

ever that the solutions obtained in [1, 61] are low-biased estimates. Hence it is not

clear whether many of our results, which tend to be higher than those of [1, 61],

are high-biased or not. It would be useful in this case to have results from a dual

method, such as those suggested by Rogers [64] and Haugh and Kogan [37], to

give high-biased estimates for comparison.

The timings results are difficult to compare due to differing computing re-

sources. Andersen [1] find low-biased estimates for the value of a swaption involv-

ing 16 factors in 20-60 seconds on a DEC Alpha 8400, depending on the accuracy

required. This pricing problem would be roughly equivalent to Contracts III and

IV in this chapter, which took anywhere from 50-700 seconds depending on the

volatility scenario and the number of grid points.


1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

d

Ear

ly e

x. p

rem

ium

Benchmark100 grid points500 grid points

(a) Local quadratic

1 2 3 4 5 6 7 8 9 10−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

d

Ear

ly e

x. p

rem

ium

Benchmark100 grid points500 grid points

(b) Local quadratic (cv)

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

d

Ear

ly e

x. p

rem

ium

Benchmark100 grid points500 grid points1000 grid points

(c) Nearest neighbour

1 2 3 4 5 6 7 8 9 10

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

d

Ear

ly e

x. p

rem

ium


(d) Nearest neighbour (cv)

Figure 5.4.1: Bermudan geometric average option estimates based on standard-

ised Sobol’ quadrature operators, two different interpolation operators and normal

Sobol’ grids. Circles, squares and triangles are slightly displaced on the x-axis for

clarity, and represent average results of ten experiments. Error bars give the ranges

of the results.


1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

1400

d

Tim

e (s

)

100 grid points500 grid points

(a) Local quadratic

1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

1400

d

Tim

e (s

)

100 grid points500 grid points

(b) Local quadratic (cv)

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

d

Tim

e (s

)

100 grid points500 grid points1000 grid points

(c) Nearest neighbour

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

d

Tim

e (s

)

100 grid points500 grid points1000 grid points

(d) Nearest neighbour (cv)

Figure 5.4.2: Timings for Bermudan geometric average option estimates based on

standardised Sobol’ quadrature operators, two different interpolation operators and

normal Sobol’ grids.


1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

d

Ear

ly e

x. p

rem

ium

1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Benchmark100 grid points500 grid points1000 grid points

(a) Random grid

1 2 3 4 5 6 7 8 9 100.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

d

Ear

ly e

x. p

rem

ium


(b) Random grid, cv

1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

d

Ear

ly e

x. p

rem

ium


(c) 3 random grids

1 2 3 4 5 6 7 8 9 10

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

d

Ear

ly e

x. p

rem

ium


(d) 3 random grids, cv

Figure 5.4.3: Bermudan geometric average option estimates based on standardised

Sobol’ quadrature and nearest neighbour interpolation operators and randomised

normal Sobol’ grids. Circles, squares and triangles are slightly displaced on the

x-axis for clarity, and represent average results of ten experiments. Error bars give

the ranges of the results.


1 2 3 4 5 6 7 8 9 100.12

0.14

0.16

0.18

0.2

0.22

0.24

d

Ear

ly e

x. p

rem

ium


(a) Standard

1 2 3 4 5 6 7 8 9 100.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

d

Ear

ly e

x. p

rem

ium


(b) cv

1 2 3 4 5 6 7 8 9 100.05

0.1

0.15

0.2

0.25

d

Ear

ly e

x. p

rem

ium


(c) Random grid

1 2 3 4 5 6 7 8 9 100.14

0.16

0.18

0.2

0.22

0.24

d

Ear

ly e

x. p

rem

ium


(d) Random grid, cv

1 2 3 4 5 6 7 8 9 100.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

d

Ear

ly e

x. p

rem

ium


(e) 3 random grids

1 2 3 4 5 6 7 8 9 10

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

d

Ear

ly e

x. p

rem

ium


(f) 3 random grids, cv

Figure 5.4.4: Bermudan geometric average option estimates based on standard-

ised Sobol’ quadrature and nearest neighbour interpolation operators, randomised

normal Sobol’ grids and where the integration is applied to the early exercise pre-

mium. Circles, squares and triangles are slightly displaced on the x-axis for clarity,

and represent average results of ten experiments. Error bars give the ranges of the

results.


100 200 300 400 5000

2

4

6

8

10

12

14

16

Grid points

Ear

ly E

x. P

rem

. (ba

sis

poin

ts)

(a) Contract (I)

100 200 300 400 5005

10

15

20

25

30

35

40

Grid points

Ear

ly E

x. P

rem

. (ba

sis

poin

ts)

(b) Contract (II)

100 200 300 400 500

60

80

100

120

140

160

180

Ear

ly E

x. P

rem

. (ba

sis

poin

ts)

(c) Contract (III)

100 200 300 400 50015

20

25

30

35

40

45

50

Grid points

Ear

ly E

x. P

rem

. (ba

sis

poin

ts)

(d) Contract (IV)

Figure 5.4.5: Swaption estimates for contracts I–IV and volatility scenario C, based

on antithetic Monte Carlo quadrature and nearest neighbour interpolation operators

and normal Sobol’ grids. Circles and solid lines give average results of ten exper-

iments and the benchmark, respectively, for K=8%, squares and dashed lines the

same for K=10% and triangles and dash-dot lines the same for K=12%. Error bars

give the ranges of the results. All results use control variates.


100 200 300 400 5000

2

4

6

8

10

12

14

Grid points

Ear

ly E

x. P

rem

. (ba

sis

poin

ts)

(a) Contract (I)

100 200 300 400 5000

5

10

15

20

25

30

35

40

Grid points

Ear

ly E

x. P

rem

. (ba

sis

poin

ts)

(b) Contract (II)

100 200 300 400 50020

40

60

80

100

120

140

Grid points

Ear

ly E

x. P

rem

. (ba

sis

poin

ts)

(c) Contract (III)

100 200 300 400 50010

15

20

25

30

35

40

Grid points

Ear

ly E

x. P

rem

. (ba

sis

poin

ts)

(d) Contract (IV)

Figure 5.4.6: Swaption estimates for contracts I–IV and volatility scenario D, based

on antithetic Monte Carlo quadrature and nearest neighbour interpolation operators

and normal Sobol’ grids, and using an inner control variate. Circles and solid lines

give average results of ten experiments and the benchmark, respectively, for K=8%,

squares and dashed lines the same for K=10% and triangles and dash-dot lines the

same for K=12%. Error bars give the ranges of the results. All results use control

variates.


100 200 300 400 5000

100

200

300

400

500

600

Grid points

Tim

e (s

)

Contract IContract IIContract IIIContract IV

(a) Volatility scenario C

100 200 300 400 5000

100

200

300

400

500

600

700

800

Grid points

Tim

e (s

)Contract IContract IIContract IIIContract IV

(b) Volatility scenario D

Figure 5.4.7: Timings for Bermudan swaption estimates based on quadrature and

nearest neighbour interpolation operators and normal Sobol’ grids. The timing

results for contract III with 500 points are omitted due to shortage of memory in

both cases.

Chapter 6

Convergence Results

6.1 Introduction

Recent work within the mathematical finance community has seen the develop-

ment of several algorithms aimed at solving high-dimensional optimal stopping

problems. Such algorithms may be used for the pricing and hedging of American-

and Bermudan-style derivative securities whose intrinsic value may be based on

a large numbers of factors. Important practical examples of such derivatives are

Bermudan swaptions, for which the outstanding notional amount worldwide runs

into the trillions of euros, and real options, which constitute a growing research

area of considerable practical importance.

Algorithms for pricing high-dimensional American options have largely been

developed in a path simulation context, starting with the works of Tilley [70] and

Barraquand and Martineau [4]. The ensuing methods have included the stochastic

tree and stochastic mesh approaches of Broadie and Glasserman [18, 19], in which

confidence intervals are constructed using high- and low-biased estimators, and the

regression-based methods of Carriere [20], who uses nonparametric techniques to

estimate the optimal stopping region, and Tsitsiklis and Van Roy [71] who use re-

gression in the context of value iteration. Longstaff and Schwartz [50] showed that

Carriere’s method could succeed in a high-dimensional setting using parametric

regressions onto a carefully chosen set of basis functions.

The new algorithms proposed in Chapters 2–4 (also published as [8, 7, 9, 10])

constitute a fundamentally different approach in that they approximate the con-

125

126 Chapter 6. Convergence Results

tinuous problem using a quasi-random or low distortion state-space discretisation,

followed by time stepping. They do not thus use path simulations as a basis for esti-

mating the continuation value, but instead employ an approximating Markov chain

to represent the dynamics in a manner which is amenable to computation. Alter-

natively, one can see these methods as irregular grid versions of finite difference

methods for partial differential equations (PDEs). Chapters 2 and 4 demonstrate

the convergent behaviour of the respective algorithms experimentally, the estimates

for ten-dimensional problems in the latter agreeing closely with benchmarks using

100,000 points and a control variate. No formal proof of convergence is provided

in those chapters.

In a similar setting, Glowinski et al. [35] prove convergence of numerical sch-

emes for variational inequalities. They make a rather strong coercivity assumption

which, as noted by Zhang [76, 77], is not satisfied in general by the Black-Scholes

operator. The operator rather satisfies the weaker Garding inequality assumption,

which Zhang uses to prove convergence of a numerical scheme for pricing Amer-

ican options on jump-diffusion processes, and based on a localised regular grid in

one dimension. Jaillet et al. [42] also appeal to the Glowinski et al. framework

to prove convergence of the Brennan-Schwartz algorithm, which amounts to an

explicit finite difference method, in one dimension.

The irregular grid scheme differs from other implementations for American op-

tion pricing, such as those of Jaillet et al. [42] and Zhang [76], in two ways. First,

the irregular grid scheme does not assume, a priori, a specification of the grid struc-

ture. This is a useful property in a high-dimensional space because, for a regular

grid, the curse of dimensionality implies an exponential increase of the number of

grid points with an increasing number of stochastic factors. The ability to use an

unstructured grid thus allows specifications for which the computation of an ap-

proximate solution is a tractable operation, even if the error related to this solution

may still be subject to the curse of dimensionality. The second difference is that

no localisation is performed before discretisation; instead, the irregular grid frame-

work considers state space discretisations which become dense in the state space as

the discretisation parameter tends to infinity. This means that convergence should

depend on only two parameters: the time step and the state space discretisation

parameter.

Proof of convergence for the irregular grid scheme is carried out in this work

6.2. Formulation 127

using the variational inequality framework of Glowinski et al. [35]. We relax the

coercivity assumption to a Garding inequality assumption, as in Zhang [76], but

in contrast to the latter we treat the multidimensional case and use a discrete ver-

sion of Gronwall’s inequality to establish stability of the approximate solutions.

The discrete Gronwall inequality leads to a better understanding of the stability

behaviour through more explicit expressions for the stability bounds.

Finally, we investigate how the convergence assumptions are satisfied in the lo-

cal consistency approach to irregular grid solutions presented in Chapter 4. Given

the inherent complexity of operator approximations on irregular grids, we are un-

able to offer an analytic investigation of this; indeed, the convergence assumptions

are related to the eigenvalues of the operator approximations, which are difficult

to analyse. Instead we provide experimental evidence that certain conditions for

stability are satisfied.

The remainder of this chapter is arranged as follows. In Section 6.2 we provide

formulations of the mathematical problem to be solved. In Section 6.3, we review

the local consistency approach to approximating process dynamics on an irregular

grid. We then provide a proof of convergence in Section 6.4 and in Section 6.5 we

investigate satisfaction of the sufficient conditions for convergence in the case of

the local consistency approach. Finally Section 6.6 draws conclusions.

6.2 Formulation

6.2.1 Stopping time formulation

We consider processes of the form

dX(t) = µ(X(t), t) dt+ σ(X(t), t) dW (t) (6.2.1)

on the time interval [0, T ] where the initial value X(0) is known almost surely,

and µ and σ are measurable with respect to the natural filtration of dW (t). For

example, in option pricing problems we would set µ to be the risk-neutral drift,

and σ the instantaneous volatility.

An optimal stopping problem on the process (6.2.1) involves finding the value

v(x0, 0) = supτ∈T

EQx0

[ψ(X(τ), τ)] (6.2.2)


where T is the time horizon, T is the set of stopping times on [0, T ] with respect

to the natural filtration, the initial value is X(0) = x0 and ψ(X, τ) is the value

achieved by stopping in state X at time τ . The expectation is taken with respect to

the process (6.2.1), which is assumed to be the risk neutral process. When applied

to problems where discounting is applicable, we assume that this is included in ψ,

that is, all values are in time zero euros.

Additionally, one may consider the numerical problem of finding the optimal

stopping time itself

τ = argsupτ∈T EQx0

[

ψ(X(τ), τ)]

, (6.2.3)

which may be useful for example in finding a low-biased estimate of the solution

to (6.2.2).

6.2.2 Variational inequality formulation

Demonstrating the convergence of solution schemes for optimal stopping prob-

lems is often simplified by appealing to the variational inequality formulation of

the problem. The connection between the two problem classes is established in

Chapter 3 of Bensoussan and Lions [5], and results concerning the convergence of

numerical schemes may be found in Glowinski et al. [35].

In contrast to the optimal stopping formulation for the value function, the vari-

ational inequality formulation is usually treated in reverse time, with the option

payoff as initial condition. In the following, we work in reverse time to maintain

consistency with [35] and with the usual analysis of initial boundary value prob-

lems.

We define an operator A giving the diffusion of the process; the construction

of A from the coefficients of (6.2.1) is given in [5]. For example, in the case of the

multidimensional extension of the Black-Scholes model we have

A = −12

d∑

i,j=1

σij(x, t)∂2

∂xi∂xj+ 1

2

d∑

i=1

σii(x, t)∂

∂xi(6.2.4)

where we assume that the risk-free rate is zero as mentioned above.


As in [42], we introduce the weighted Sobolev spaces

Wm,p,µ(Rd) =

v ∈ Lp(

Rd, e−µ|x|)

: Dαv ∈ Lp(

Rd, e−µ|x|)

, |α| ≤ m

(6.2.5)

where Dα is the partial derivative corresponding to the multiindex α. We let Hµ =

W 0,2,µ and Vµ = W 1,2,µ; further we denote the Hµ norm and inner product by

| · |µ, (·, ·)µ and the Vµ norm and inner product by ‖ · ‖µ, ((·, ·))µ. So that

|v|2µ =

∫

Rd

v(x)2e−µ|x|dx (6.2.6)

‖v‖2µ = |v|2µ +

∑

|α|=1

|Dαv|2µ . (6.2.7)

The introduction of weighted spaces facilitates the consideration of intrinsic

functions ψ which may not be integrable in a non-weighted space. This is the case

with call options for example.

The variational inequality equivalent to the optimal stopping problem (6.2.2) is

to find v ∈ Hµ such that

v(·, T ) ≡ ψ(·, T )

v(x, s) ≥ ψ(x, s)

(

∂v

∂t−Av, u− v

)

µ

≤ 0 a.e. ∀u ≥ ψ, t ∈ [0, T ]

(6.2.8)

for (x, s) ∈ Rd × [0, T ].

6.2.3 General variational inequality formulation

In Section 6.3 we will consider numerical methods for the solution of general para-

bolic time-dependent inequalities of type I, as defined in Chapter 6 of Glowinski et

al. [35]. We follow the setting of [35], except that we assume a Garding inequality

for the bilinear form in place of strict coercivity.

We employ two Hilbert spaces H and V , with respective inner products (·, ·)

and ((·, ·)), respective norms | · | and ‖ · ‖, and where V is a dense subset of H with

continuous injection. Typically H will be an L2 space, and V the Sobolev space

of degree one, i.e. where all first order derivatives are in H .


Example 6.2.1 As mentioned in Section 6.2.2, it will often be advantageous to

work with weighted spaces so that the obstacle or payoff remains integrable. Hence,

for such problems we may use H = Hµ and V = Vµ as defined above.

From the above continuity requirement we can conclude that, for v ∈ V

|v| ≤ c‖v‖ (6.2.9)

for some constant c. We denote by V ′ the dual of V with respect to H , thus

obtaining the Gelfand triple

V → H → V ′ (6.2.10)

with dense embeddings.

To accommodate the constraints we introduce a convex set K ⊂ V in which

the solution should lie for each t, and we assume the initial condition v0 ∈ K . We

consider a linear operator A : V → V ′ (for example (6.2.4)) and its associated

bilinear form a(v, v) = (Av, v) which satisfies a Garding inequality

a(v, v) + ρ|v|2 ≥ α‖v‖2 (6.2.11)

for some α > 0, ρ ≥ 0. In the case where ρ = 0 and α > 0, this becomes a strict

coercivity condition. As noted in Zhang [76], the Black-Scholes operator does not

in general satisfy a strict coercivity condition.

In order to fully define the problem, we must define the dual norm and intro-

duce some abstract spaces. For a full justification of the following steps the reader

is referred to Glowinski et al. [35]. We first introduce a space for the solutions of

the time-dependent problem. V ′ is first equipped with the norm

‖w‖∗ = supv∈V,‖v‖=1

(w, v) (6.2.12)

and we define the Hilbert space and subspace

W ([0, T ]) =

v ∈ L2([0, T ];V ) :dv

dt∈ L2([0, T ];V ′)

(6.2.13)

W0([0, T ]) =

v ∈W ([0, T ]) : v(0) = v0 ∈ H

(6.2.14)


where L2([0, T ];V ) denotes the space of functions t→ f(t) which are measurable

from [0, T ] → V such that

‖f‖L2([0,T ];V ) =

(∫ T

0‖f(t)‖2

V dt

)1/2

< +∞. (6.2.15)

Finally we introduce the convex spaces

K =

v ∈W ([0, T ]) : v(t) ∈ K a.e. [0, T ]

(6.2.16)

K0 =

v ∈W0([0, T ]) : v(t) ∈ K a.e. [0, T ]

. (6.2.17)

The problem to be solved is then to find v ∈ K0 such that

∫ T

0

(

∂v

∂t+ Av − f, u− v

)

dt ≥ 0 ∀u ∈ K (6.2.18)

for some appropriate L2 function f (the zero function in our case), or equivalently

(

∂v

∂t+ Av − f, u− v

)

≥ 0 ∀u ∈ K, a.e. [0, T ]. (6.2.19)

Remark 6.2.1 The problem formulation introduced accommodates a large class

of parabolic variational problems, of which optimal stopping problems are just one

example.

Example 6.2.2 We recall the American option pricing problem from Example

6.2.1, which is an initial value problem in reverse time in the current setting. We

let

K = Kµ = v ∈ Vµ : v ≥ ψ ⊂ Vµ. (6.2.20)

In this case we set v0 = ψ ∈ Kµ.

Assuming the covariance matrix Σ = (σij) is symmetric positive definite, we

can set α ∈ (0, 14λ) in (6.2.11) where λ is the smallest eigenvalue of Σ. This can


be seen as follows, using the operator defined in (6.2.4):

a(v, v) = (Av, v)

= −12

∫

∑

i,j

σij∂2v

∂xi∂xjv −

∑

i

σii∂v

∂xiv

e−µ|x|dx

= 12

∫

∑

i,j

σij∂v

∂xi

∂v

∂xj+∑

i

σii − µ∑

j

xj

|x|σij

∂v

∂xiv

e−µ|x|dx

≥ 12

∫

(∇v)′Σ(∇v) +∑

i

σii − µ∑

j

σij

∣

∣

∣

∣

∂v

∂xi

∣

∣

∣

∣

|v|

e−µ|x|dx

≥ 2α∑

i

∣

∣

∣

∣

∂v

∂xi

∣

∣

∣

∣

2

µ

− 12

∑

i

∣

∣

∣

∣

σii − µ∑

j

σij

∣

∣

∣

∣

∣

∣

∣

∣

∂v

∂xi

∣

∣

∣

∣

µ

|v|µ .

Now

a(v, v) + ρ|v|2µ

≥ 2α∑

i

∣

∣

∣

∣

∂v

∂xi

∣

∣

∣

∣

2

µ

− 12

∑

i

∣

∣

∣

∣

σii − µ∑

j

σij

∣

∣

∣

∣

∣

∣

∣

∣

∂v

∂xi

∣

∣

∣

∣

µ

|v|µ + ρ|v|2µ

= α‖v‖2µ +

∑

i

α

∣

∣

∣

∣

∂v

∂xi

∣

∣

∣

∣

2

µ

+ρ− α

d|v|2µ − 1

2

∣

∣

∣

∣

σii − µ∑

j

σij

∣

∣

∣

∣

∣

∣

∣

∣

∂v

∂xi

∣

∣

∣

∣

µ

|v|µ

.

It is now possible to choose ρ large enough such that each term in the sum is

positive, thus showing that the bilinear form satisfies a Garding inequality.

6.3 Solution framework

We now propose a solution framework for solving variational inequalities in a gen-

eral Hilbert space setting. Numerical schemes for solving the variational inequality

(6.2.8) can be developed in this setting, for which relevant examples will be pre-

sented. The setting follows that of Glowinski et al. [35], except that we do not

assume strict coercivity of the diffusion operator or its discrete counterpart.

The framework involves the following steps:

6.3. Solution framework 133

1. discretisation of space and time,

2. approximation of constraints and operators in the discretised setting,

3. numerical solution of the discretised problem.

In this section we will often reuse a constant c for various different bounds and

continuity constants. Convergence results will be presented in Section 6.4.

6.3.1 Discretisation of space and time

We consider separate discretisations of space and time. Such discretisations are

convenient for solving initial value problems, since time stepping methods allow

division of the problem into a sequence of smaller problems. Having a constant

grid in the state space also allows implicit solutions to be more easily considered.

For the space discretisation, we first introduce the auxiliary Hilbert space Φ

to be used in relating V to its discretisation; Φ in effect is intended to contain the

extra information required to know whether for some v ∈ H we also have v ∈ V .

In particular, when H = W 0,2,µ and V = W 1,2,µ as defined in (6.2.5), Φ would

correspond to first order derivatives. We now denote

F = H × Φ (6.3.1)

and the corresponding extension isomorphism

σ : V → F (6.3.2)

which we will always take to be the identity plus first derivative information

σv =

(

v,∂v

∂x1, . . . ,

∂v

∂xd

)

. (6.3.3)

Consider now finite dimensional Hilbert spaces Hn and Vn approximating H

and V respectively where n denotes the dimension of the spaces. The properties

required for these approximations are now studied. We denote the respective norms

of Hn and Vn by | · |n and ‖ ·‖n, the respective inner products by (·, ·)n and ((·, ·))n,

and introduce linear extension operators

qn : Vn → H (6.3.4)

pn : Vn → F (6.3.5)


relating elements of the discrete Hilbert spaces to those of the continuous spaces,

where we assume for v ∈ Vn

|qnv| = |v|n (6.3.6)

‖pnv‖ ≤ c‖v‖n (6.3.7)

where c does not depend on n. These operators are related in that pnv = (qnv, . . .)

and pn is assumed to be convergent in that for each v ∈ V , there exists a bounded

sequence vn ∈ Vn such that

pnvn → σv strongly in F . (6.3.8)

The following relationships are assumed between the norms in the finite di-

mensional spaces for some constant c and (increasing) function s(n):

|v|n ≤ c‖v‖n (6.3.9)

‖v‖n ≤ s(n)|v|n. (6.3.10)

Note that the first inequality is the discrete version of (6.2.9). The function s(n),

which in general increases without bound as n → ∞, will appear in stability and

convergence conditions; we will see in particular that specifying s(n) to be as small

as possible leads to more favourable estimates.

Example 6.3.1 Consider a generic grid, or set of states

Xn = x1, . . . , xn (6.3.11)

where Xn becomes dense in Rd as n → ∞. For example, one may consider

generating random or quasi-random states from Rd according to an appropriate

density. A regular grid may also be used, although in this case either the problem

must first be localised or the span of the grid must increase while the resolution

becomes simultaneously finer.

We now proceed to build an approximating Hilbert space based on indicator

functions of the Voronoi cells of Xn. Define the ith Voronoi cell of a grid X as the

set

CX (i) =

x ∈ Rd : |x− xi| = minj

|x− xj |

(6.3.12)


−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Figure 6.3.1: Voronoi cells of a grid X in dimension 2 containing 20 points. Dis-

tances are with respect to the Euclidean norm and dashed lines continue to infinity.

and the Voronoi index of x ∈ Rd with respect to X as the positive integer

JX (x) = argminj |x− xj|. (6.3.13)

Further, define the (possibly multivalued) k-nearest neighbour mapping for k ≤ n

as

NX ,k(x) =

j1, . . . , jk : j1 =argminj |x− xj|, j2 = argminj 6=j1 |x− xj|,

. . . , jk = argminj /∈j1,...,jk−1|x− xj|

,

(6.3.14)

giving the ordered indices of the k nearest neighbours of x in X . Figure 6.3.1

provides an example of the Voronoi cells for a grid X containing 20 points.

We now introduce the Hilbert spaces

Hn = Vn = RX

∼=

v ∈ L2 : v(x) =n∑

i=1

ciICX (i)(x), ci ∈ R

(6.3.15)

with norms to be specified, and the isomorphically equivalent space given will

form the image of the extension operators qn and pn. We see that this image is not

contained in V ; the extension operator pn will thus constitute an exterior approx-

imation as noted in [35]. On the other hand, an interior approximation may result

for example from using an extension operator related to a finite element discretisa-

tion.


Let us now define the extension operator qn in the obvious manner for v ∈ Hn

[qnv](x) = vJX (x). (6.3.16)

This leads us through condition (6.3.6) to the norm definition for Hn

|v|2n =

∫

[qnv](x)2e−µ|x|dx

=

n∑

i=1

θiv2i (6.3.17)

where

θi =

∫

CX (i)e−µ|x|dx. (6.3.18)

The space Hn is thus a weighted discrete L2 space where the weights are decreas-

ing with the norm of xi, but increasing with the size of the Voronoi cell containing

xi.

In this setting the operator pn should give information about the derivatives of

v ∈ Vn. We set

[pnv](x) = ([qnv](x), [δ1v](x), . . . , [δdv](x)) (6.3.19)

where the discrete derivative operators δi are defined at x ∈ Rd through using first

order Taylor series expansions at the d + 1 nearest neighbours of x as follows: let

(j1, . . . , jd+1) be the (ordered) d + 1 nearest neighbours of xi, we then solve the

equations

v(xj1) = v(xi) +d∑

j=1

(xj1 − xi)∂v

∂xj(xi)

...

v(xjd+1) = v(xi) +

d∑

j=1

(xjd+1− xi)

∂v

∂xj(xi)

for the ∂v∂xj

, which leads us to define

[δiv](x) =(

vj1 · · · vjd+1

)

A−1x ei+1 (6.3.20)


where

Ax =

(

1 · · · 1

(xj1 − x) · · · (xjd+1− x)

)

(6.3.21)

and ei is the ith canonical unit vector. The norm in Vn is defined analogously:

‖v‖2n = |v|2n +

d∑

i=1

|δiv|2n. (6.3.22)

This allows us to set c = 1 in (6.3.7) and (6.3.9). To find a suitable bound s(n)

in (6.3.10), we consider the worst-case effect of the matrices Ax in (6.3.20). We

propose the bound

s(n) = 1 + dλ(n)−2η (6.3.23)

where η is the greater of the maximum number of neighbours per point and the

maximum number of times a point is a neighbour (with respect to the derivative

estimates), and λ(n) is the minimum λj where, for each j, λ−2j is the smallest

eigenvalue of AjA′j where Aj ≡ Axj

as defined above. We demonstrate the valid-

ity of the bound (6.3.23) as follows:

|δiv|2n =

n∑

j=1

(

(

vj1 · · · vjd+1

)

A−1j ei+1

)2θj

=n∑

j=1

(

d+1∑

k=1

vjke′kA

−1j ei+1

)2

θj

≤n∑

j=1

d+1∑

k=1

v2jkλ−2

j θj

≤ λ−2η|v|2n.

In the third line we have used the following matrix inequalities∣

∣

∣ekA

−1j ei+1

∣

∣

∣≤ ‖A−1

j ‖∆

≤ ‖A−1j ‖2

=

(

max eigenvalue of

(

(

A−1j

)′A−1

j

))1/2

=(

min eigenvalue of (A′jAj)

)−1/2

=: λ−1j


where we use the matrix norm ‖A‖∆ = maxi,j |aij |. The inequality ‖A‖∆ ≤

‖A‖2 can be seen as follows:

‖A‖∆ = |aij | for some i, j

≤

(

∑

k

a2kj

)1/2

=‖Aej‖2

‖ej‖2≤ max

x6=0

‖Ax‖2

‖x‖2= ‖A‖2.

Note that one could use other matrix norms than ‖A−1j ‖2 in the estimation; however

these other norms are somewhat more difficult to interpret in terms of the properties

of Aj itself.

Finally

‖v‖2n = |v|2n +

d∑

i=1

|δiv|2n

≤ |v|2n + dλ(n)−2η|v|2n

=(

1 + dλ(n)−2η)

|v|2n. (6.3.24)

We thus see that the factor s(n) =(

1 + dλ(n)−2η)1/2

bounding the Vn norm

involves the dimension d, the maximum number of neighbours η used for the

derivative estimations and the term λ(n) which is a function of the neighbour dis-

tances and their conditioning.

For example, for a regular grid in one dimension X = x1, x1 + δx, . . . , xn,

and using the δi as defined in (6.3.20), we have for each j = 2, . . . , n− 1

Aj =

(

1 1

0 ±δx

)

. (6.3.25)

Assuming δx < 1, we have λ2j ∈ ρ(A′

jAj) = 1 + 12δx

2 ±(

1 + 14h

4)1/2

'

12δx

2, 2. Alternatively, for the δi proposed in [35] and [76] we have

Aj =

(

1 1

−12δx

12δx

)

, (6.3.26)

in which case λ2j ∈ ρ(A′

jAj) = 12δx

2, 2 (the same as above up to O(δx4)), and

λ(n)−2 = 2/δx2. Noting that in this case η = 2, we have

s(n) =

(

1 +4

δx2

)1/2

(6.3.27)


as found in [76]. Note that s(n) depends on n through δx = (xn − x1)/(n− 1).

We assume that time is discretised using K steps

t = t0 < · · · < tK = T (6.3.28)

and denote the time step sizes by δtk = tk+1 − tk for k = 0, . . . ,K − 1. As

K → ∞, we require

maxk

δtk → 0. (6.3.29)

Given approximate solutions v(n)k (x) corresponding to an n-dimensional space

discretisation and t = tk, k = 0, . . . ,K , we consider the (timewise) piecewise

constant extension

v(n,δt)(x, t) =

K∑

k=0

v(n)k (x)11(k = argminj |t− tj |). (6.3.30)

Example 6.3.2 The regularly spaced discretisation

tk =k

KT k = 0, . . . ,K (6.3.31)

is often the simplest to work with, although it cannot be expected that this discreti-

sation leads to an optimal convergence rate in general. For example, the time step

size in certain numerical solution algorithms for PDEs is determined adaptively

depending on the shape of the computed solutions.

6.3.2 Approximation of constraints and operators

We now discuss approximation of the constraints and operators in the discrete set-

ting introduced in Section 6.3.1. For operators we refer to the partial differential

operators ∂∂t and A, and for constraints we refer to the convex spaces K in (6.2.19).

Let us first consider approximation of the constraints. The requirement v ∈ K

for each time t is approximated by the requirement

v(n)k ∈ Kn ⊆ Vn (6.3.32)


for each time index k = 1, . . . ,K where Kn is closed and convex. The Kn are

assumed to be consistent in that, if for vn ∈ Kn, pnvn → ξ ∈ F weakly then

ξ ∈ σK . We assume further that Kn are convergent in that for all v ∈ K there

exists a bounded sequence vn ∈ Kn such that pnvn → σv ∈ F strongly.

Note that the extension of Kn to H , based on the extension operators pn and

qn, is not in general required to be a subset of K , or even to be contained in V

itself. In particular this cannot be expected in the case of exterior approximations

(see page 135).

Example 6.3.3 In the case of American options, following from Example 6.3.1, a

simple specification of Kn is

Kn = v ∈ Vn : v ≥ ψn (6.3.33)

where ψn(x) = ψ(xJXn (x)) and Xn = x1, . . . , xn is the nth grid.

We now consider the operator ∂∂t , for which we specify a standard first order

approximation for tk, k = 0, . . . ,K − 1:

∂v

∂t(x, tk) '

1

δtk

(

v(n,k)(x, tk+1) − v(n,k)(x, tk))

. (6.3.34)

This definition may be extended to t ∈ [0, T ] if required; however we shall only

require approximations at times tk.

We finally consider an approximation An to the operator A in the discretisa-

tion introduced above and the corresponding bilinear form an(u, v) = (Anu, v)n.

As in the continuous problem, the approximation is assumed to satisfy a Garding

inequality

an(v, v) + ρ|v|2n ≥ α‖v‖2n (6.3.35)

for some constants α > 0, ρ ≥ 0 not depending on n. These constants are not in

general the same as those found in the continuous Garding inequality formulation

(6.2.11).

The bilinear form is also assumed to be continuous in that

|an(u, v)| ≤ c‖u‖n‖v‖n (6.3.36)


for all u, v ∈ Vn where c > 0 is a constant independent of n. Further, the bilinear

form is assumed to converge in the following sense: suppose pnvn → σv weakly

in F and pnwn → σw strongly in F , then an(vn, wn) → a(v, w), an(wn, vn) →

a(w, v) and lim infn an(vn, vn) ≥ a(v, v).

Example 6.3.4 For American option pricing problems, which constitute varia-

tional inequalities defined on an unbounded domain, most authors perform a lo-

calisation before discretising. The localised domain is assumed to be large enough

to guarantee a suitably small truncation error. For a regular rectilinear grid on a lo-

calised domain, a sequence An satisfying the above conditions can be constructed

using standard finite difference methods, as shown in Glowinski et al. [35] and

Zhang [76]. In particular, no weighting is needed in the case of a localised domain

to make a call payoff integrable.

For multidimensional problems, such a rectilinear domain suffers from the

curse of dimensionality, and may not be appropriate from importance sampling

considerations. For these reasons, irregular grid methods have been suggested in

Chapters 2–4 (also published as [8, 7, 9, 10]) for the solution of multidimensional

problems. Two stable constructions of An are suggested in those chapters, namely

the method of Chapter 2 (also [8, 7, 9]) involving the logarithm or root of a tran-

sition matrix corresponding to importance sampling weights and the method of

Chapter 4 (also [10]) involving the solution of a large number of small linear pro-

gramming problems. Given the unbounded domain and the fact that these methods

do not involve localisation, one must consider weighted norms in the discretisation

in order to allow non-L2 payoffs, as suggested in Example 6.3.1.

We thus form the weighting matrix Θ = diag(θ1, . . . , θn) where the weights

θi are given in (6.3.18), and consider the weighted bilinear form

an(u, v) = (Anu, v)n = v′ΘAnu. (6.3.37)

In Section 6.5 we will investigate how the conditions on an are satisfied for the

local consistency method as proposed in Chapter 4.

Finally, we assume that f is continuous and approximated in a (timewise) step-


wise constant manner

f (n,k)(x, t) =

K∑

k=0

f(n)k (x)11(k = argminj |t− tj |) (6.3.38)

where f (n)k (x) = f(tk, x). We are now in a position to form the system of equa-

tions whose solution approximates that of (6.2.19):(

vi+1n − vi

n

k+Anv

i+θn , u− vi+1

n

)

n

≥ 0 vi+1n ∈ Kn, ∀u ∈ Kn (6.3.39)

for i = K − 1, . . . , 0 where v0n = ψn and vi+θ

n = (1 − θ)vin + θvi+1

n is the θ-

weighted value approximation for the ith iteration. In the case where Kn = v ∈

Vn : v ≥ ψ for some obstacle ψ, the discrete variational inequality (6.3.39) may

be reformulated as a sequence of linear complementarity problems

0 ≤ vi−1 − ψi−1 ⊥MLvi−1 −MRv

i ≥ 0 (6.3.40)

for i = K−1, . . . , 0 where v0n = ψn,ML = I−Anθδt andMR = I+An(1−θ)δt.

6.4 Convergence of the discretised problems

Having discussed the functional setting of the discretised problem and proposed

conditions for convergence in Section 6.3, we now demonstrate the convergence of

the discretised solutions (6.3.39) to the true solutions (6.2.19).

In Section 6.4.1 we establish conditions under which the solution to the discre-

tised system (6.3.39) is stable using an M -matrix assumption. Using the Garding

inequality assumption we prove in Section 6.4.2 the stability of the solutions along

with some related quantities. We then place this in the framework of Glowinski et

al. to achieve a theorem of convergence.

For simplicity we now assume a constant time step, although the following can

be easily modified to accommodate a variable time step.

6.4.1 Stability under M -matrix assumption

We now provide conditions under which the solutions of the discretised variational

inequalities (6.3.39) are stable. This is done through the complementarity repre-

sentation of the variational inequalities.

6.4. Convergence of the discretised problems 143

We remind the reader of some matrix classes which will be used in the follow-

ing analysis. For a full treatment of matrix classes we refer the reader to Berman

and Plemmons [6] or Cottle, Pang and Stone [23].

Definition 6.4.1 A real square matrix is said to be a Z-matrix if its off-diagonal

entries are nonpositive.

Definition 6.4.2 A real square matrix is said to be an M -matrix if it is a Z-matrix

with nonnegative diagonal entries.

Definition 6.4.3 A real square matrix is said to be a P -matrix if all of its principal

minors are positive.

The explicit case

The complementarity problems are

0 ≤ vi−1 − ψi−1 ⊥ vi−1 −Mvi ≥ 0 (6.4.1)

for i = K, . . . , 1. The payoff functions ψi are given and are allowed to vary over

time. The matrix M is formed from the infinitesimal generator as follows

M = (1 − rδt)I +Aδt (6.4.2)

and has the property of being a P -matrix and having row sums 1 − rδt. In the

explicit case the complementarity problems reduce to the simpler problem

vi−1 = max(

ψi−1,Mvi)

(6.4.3)

where max gives the pointwise maximum.

Lemma 6.4.1 Suppose thatA is anM -matrix with zero row sums and r ≥ 0. Then

under the stability condition

δt ≤1

‖A− rI‖∆(6.4.4)

where ‖A‖∆ = maxi,j|aij|, the solution at time index i = 0 of the explicit system

of complementarity problems (6.4.1) satisfies

‖v0‖∞ ≤ maxi=0,...,K

(1 − rδt)i‖ψi‖∞. (6.4.5)


Proof. From (6.4.3) we have

‖vi−1‖∞ ≤ max(

‖ψi−1‖∞, ‖Mvi‖∞)

≤ max(

‖ψi−1‖∞, ‖M‖∞‖vi‖∞)

= max(

‖ψi−1‖∞, (1 − rδt)‖vi‖∞)

.

Applying this inequality recursively with vK ≡ ψK gives the required result.

The stability bound on δt is presented in terms of the maximum absolute entry

of the matrix M . We now investigate this bound and its asymptotic properties.

Lemma 6.4.2 Suppose that A is an M -matrix with zero row sums and r ≥ 0, and

that A is locally consistent for a process having covariance matrix Σ = (σij) (de-

note σ2i ≡ σii). Suppose further that the Euclidean distances between connected

points in the grid X lie in the interval [ε1, ε2]. Then the norm appearing in Lemma

6.4.1 is bounded by

1

ε22

∑

σ2i + r ≤ ‖A− rI‖∆ ≤

1

ε21

∑

σ2i + r. (6.4.6)

Proof. From the feasibility conditions we have

aij1δx2iji1 + · · · + aijηδx

2ijη1 = σ2

1

...

aij1δx2ijid

+ · · · + aijηδx2ijηd = σ2

d.

The hypothesis regarding the distance between connected points implies that, for

each δxijk,

ε21 ≤ δx2ijk1 + · · · + δx2

ijkd ≤ ε22.

Summing the above equations and applying the inequality gives

ε21∑

k

aijk≤∑

σ2i ≤ ε22

∑

k

aijk. (6.4.7)

Now the off-diagonal entries of A − rI are exactly the aijkin (6.4.7), which

are nonnegative and satisfy the bounds

1

ε22

∑

σ2i ≤ |aijk

| ≤1

ε21

∑

σ2i .


The diagonal entries of A− rI are −∑

aijk− r, which are equal in absolute value

to∑

aijk+ r, provided r ≥ 0, hence they satisfy the bounds

1

ε22

∑

σ2i + r ≤

∣

∣

∣−∑

aijk− r∣

∣

∣ ≤1

ε21

∑

σ2i + r.

Putting these bounds together completes the proof.

Lemmas 6.4.2 and 6.4.1 are now combined to give a stability condition in terms

of the ratio between the time step and the minimum point separation ε1.

Lemma 6.4.3 Under the conditions of Lemma 6.4.2, the stability condition in

Lemma 6.4.1 holds providedδt

ε21≤

1 − rδt∑

σ2i

. (6.4.8)

It is clear that δt/ε2 must be less than 1/∑

σ2i to guarantee stability as δt→ 0.

The fully implicit case

The stability condition in the explicit case can be rather restrictive, especially in a

low dimension where the point separation decreases more rapidly with grid size.

Since implicit methods often exhibit greater stability, we now investigate their

properties.

The complementarity problems are now

0 ≤ vi−1 − ψi−1 ⊥Mvi−1 − vi ≥ 0 (6.4.9)

for i = K, . . . , 1. Again the payoff functions ψi are given and are allowed to vary

over time. The matrix M is now given by

M = (1 + rδt)I −Aδt. (6.4.10)

Lemma 6.4.4 Suppose thatA is anM -matrix with zero row sums and r ≥ 0. Then

the solution at time index i = 0 of the system of implicit complementarity problems

(6.4.9) satisfies

‖v0‖∞ ≤ maxi=0,...,K

(1 + rδt)i‖ψi‖∞. (6.4.11)


Proof. Consider the vector

v = max(

(1 + rδt)−1‖vi‖, ‖ψi−1‖∞)

11. (6.4.12)

We claim that v is feasible for (6.4.9), though it does not necessarily satisfy the

complementarity conditions. For the first inequality in (6.4.9) this is obvious. For

the second we have, recalling that A has zero row sums,

Mv − vi = (1 + rδt)v − vi

= max(

‖vi‖, (1 + rδt)‖ψi−1‖∞)

11 − vi

≥ 0.

The solution v being feasible but not necessarily satisfying the complemen-

tarity conditions implies, according to Theorem 3.11.6 in [23] and noting that an

M -matrix is also a Z-matrix, that v dominates (pointwise) the true solution v i−1.

Thus

‖vi−1‖∞ ≤ ‖v‖∞

= max(

(1 + rδt)−1‖vi‖, ‖ψi−1‖∞)

.

Applying this inequality recursively with vK ≡ ψK gives the desired result.

Remark 6.4.1 The fully implicit method is thus unconditionally stable, unlike the

explicit method where a stability condition was imposed in Lemma 6.4.3. We shall

see shortly that θ case time stepping methods admit a similar stability condition

to the explicit case. The implicit method is thus the only first order time stepping

scheme we can prove to be unconditionally stable, as also found in Chapter 6 of

Glowinski et al. [35].

The θ case

The θ case lies “between” the explicit and implicit problems. The complementarity

problems are

0 ≤ vi−1 − ψi−1 ⊥MLvi−1 −MRv

i ≥ 0 (6.4.13)


for i = K, . . . , 1. Again the payoff functions ψi are given and are allowed to vary

over time. The matrices ML and MR are given by

ML = (1 + rθδt)I −Aθδt (6.4.14)

MR = (1 + r(1 − θ)δt)I +A(1 − θ)δt. (6.4.15)

It is simple to extend the stability conditions presented in the explicit case to

the θ case. The conditions may be weaker depending on the value of θ.

Lemma 6.4.5 Suppose A is an M -matrix with zero row sums and r ≥ 0. Then

under the stability condition

δt ≤1

(1 − θ)‖A− rI‖∆(6.4.16)

where ‖A‖∆ = maxi,j|aij |, the solution at time index i = 0 of the θ-case system

of complementarity problems (6.4.13) satisfies

‖v0‖∞ ≤ maxi=0,...,K

(

1 − r(1 − θ)δt

1 + rθδt

)i

‖ψi‖∞. (6.4.17)

Lemma 6.4.6 Under the conditions of Lemma 6.4.2, the stability condition in

Lemma 6.4.5 holds provided

δt

ε2≤

1 − r(1 − θ)δt

(1 − θ)∑

σ2i

. (6.4.18)

We require that δt/ε2 must be less than 2/(1 − θ)∑

σ2i to guarantee stability

as δt → 0, which is indeed weaker than the condition for stability in the explicit

case.

6.4.2 Stability and convergence under Garding inequalityassumption

Following the framework proposed in Glowinski et al. [35], we investigate con-

vergence of the discretised variational inequalities. We refer to [35] for a detailed

treatment of the numerical solution to variational inequalities in an abstract setting.

In addition to stability, the results presented below will establish convergence

of irregular grid schemes. The stability result relates not only to the computed so-

lution, but also to the partial derivatives of its extension under pn. This is a stronger


result than that of Section 6.4.1 where only stability of the value function was es-

tablished. The convergence result relies crucially on the satisfaction of a Garding

inequality in the discretised setting. Previously Glowinski et al. [35] provided the

same result under a strict coercivity assumption (a Garding inequality with ρ = 0)

and Zhang [76] proved stability of a localised regular grid discretisation method

for pricing American options on a one-dimensional jump-diffusion process, and

under a Garding inequality assumption. Jaillet et al. [42] prove convergence of

the Brennan-Schwartz algorithm (the explicit finite difference method in one di-

mension) using the framework of [35]. Matache et al. investigate convergence of

a wavelet discretisation method for pricing European options [52] and American

options [51] on single assets following one-dimensional Levy processes.

All convergence results mentioned for American option pricing problems con-

sider only one-dimensional numerical schemes. Although these results may well

be generalised to higher dimensions, their generalisations would involve regular

grid constructions involving intractable numbers of grid points. Our investigation

of convergence for irregular grid schemes provides an extension to higher dimen-

sions which is tractable computationally. Note that the approximation error may

still be subject to the curse of dimensionality.

Our proof of stability mainly follows the arguments of Glowinski et al. [35] and

Zhang [76], except that, in contrast to the former we assume a Garding inequality

instead of strict coercivity, and in contrast to the latter we do not concentrate on

the one-dimensional case, but remain in a multidimensional setting. We also use a

discrete version of Gronwall’s lemma rather than the integral form. This allows us

to quantify the form of the error more precisely.

Theorem 6.4.1 Let ukn be the solution to the discretised variational inequality

(6.3.39) at iteration k corresponding to t = k/K = kδt. Assume that the dis-

crete Garding inequality (6.3.35) holds for some α, ρ > 0, then the quantities

vkn, pnv

kn,

k∑

i=1

|vi+1n − vi

n|2n (6.4.19)

are uniformly bounded for 0 ≤ kδt ≤ T with n and δt satisfying the stability

assumption

1 − 2δt(1 − θ)c2s(n)2

α≥ β > 0 (6.4.20)

for some β and where c is the continuity constant appearing in (6.3.36).


Remark 6.4.2 Note that the stability condition is similar to that of Lemma 6.4.6.

In one dimension, recalling from (6.3.27) that s(n) ' 1/δx2 , it is clear that each of

these conditions bounds the size of δt/δx2 and moreover the bounds both become

infinite as the implicitness parameter θ → 1.

Before proving Theorem 6.4.1, we introduce the following two lemmas. The

first establishes a discrete Gronwall inequality, and the second a limit for the esti-

mate implied by the Gronwall inequality.

Lemma 6.4.7 Suppose ai are nonnegative quantities for i ∈ N and

ak ≤ d0 + d1

k−1∑

i=0

ai + d2

k−1∑

i=0

bi, (6.4.21)

holds for all k ∈ N where bi,di are nonnegative constants. Then the inequality

ak ≤ d0(1 + d1)k−1 + d1(1 + d1)

k−1a0 + d1

k−1∑

i=0

(1 + d1)k−i−1bi (6.4.22)

holds for all k ∈ N.

Proof. We proceed to prove the inequality (6.4.22) by induction. The inequality

(6.4.22) certainly holds for k = 1. Assume now that it holds for all k ≤ m for


some m > 1. We have then for am+1

am+1 ≤ d0 + d1

m∑

i=0

ai + d2

m∑

i=0

bi

= d0 + d1a0 + d2

m∑

i=0

bi

+ d1

m∑

i=1

d0(1 + d1)i−1 + d1(1 + d1)

i−1a0 + d2

i−1∑

j=0

(1 + d1)i−j−1bj

= d0

(

1 + d1

m∑

i=1

(1 + d1)i−1

)

+ d1

(

1 + d1

m∑

i=1

(1 + d1)i−1

)

a0

+ d2

b0 +m∑

i=1

bi + d1

i−1∑

j=0

(1 + d1)i−j−1bj

= d0(1 + d1)m + d1(1 + d1)

ma0 + d2

m∑

i=0

(1 + d1)m−ibi

where the last equality can be shown using standard summation formulae.

Lemma 6.4.8 Assume that bi = b(iδt) for some function b(·) and δt > 0, and

write aδt(t) = a[t/δt] where [·] is the integer part and the ak satisfy the inequality

ak ≤ c0 + c1δtk∑

i=0

ai + c2δtk−1∑

i=0

bi (6.4.23)

for all n ∈ N. Then as δt→ 0, k → ∞ and t = kδt remaining fixed, we have

lim supk,δt

aδt(t) ≤ c0ec1t + c2

∫ t

0ec1(t−s)b(s)ds. (6.4.24)

Proof. We first reformulate (6.4.23) as

ak ≤c0

1 − c1δt+

c1δt

1 − c1δt

k−1∑

i=0

ai +c2δt

1 − c1δt

k−1∑

i=0

bi. (6.4.25)


Using the notation of Lemma 6.4.7, we set

d0 =c0

1 − c1δt, d1 =

c1δt

1 − c1δt, d2 =

c2δt

1 − c1δt

and we note that as δt→ 0

d1 → 0

1 + d1 → 1

(1 + d1)k =

(

1 −c1t

k

)−k

→ ec1t

k∑

i=0

δt(1 + d1)k−ibi →

∫ t

0ec1(t−s)b(s)ds

where the limits are taken as δt → 0, k → ∞ and holding t = kδt constant.

Applying Lemma 6.4.7, and taking the limit on the RHS of (6.4.22) gives

d0(1 + d1)k−1 + d1(1 + d1)

k−1a0 + d1

k∑

i=0

(1 + d1)k−ibi

−→ c0ec1t + c2

∫ t

0ec1(t−s)b(s)ds

as required.

Proof of Theorem 6.4.1. We first recall some identities:

vi+θ = θvi+1 + (1 − θ)vi (6.4.26)

2(a− b, a) = |a|2 − |b|2 + |a− b|2 (6.4.27)

2(a, b) ≤ 2|a||b| ≤1

ε|a|2 + ε|b|2, ε > 0. (6.4.28)

We will also make use of the discrete Garding inequality (6.3.35) and the continuity

assumption (6.3.36).

Starting from the discretised variational inequality (6.3.39), and noting that

ψn ∈ Kn, we have

1

δt(vi+1

n − vin, ψn − vi+1

n )n + an(vi+θn , ψn − vi+1

n )

≥ (f i+θn , ψn − vi+1

n )n. (6.4.29)


Now using (6.4.27) with a = vi+1n − ψn, b = vi

n − ψn, we have

(vi+1n − vi

n, ψn − vi+1n )n = |vi+1

n − ψn|2n − |vi

n − ψn|2n + |vi+1

n − vin|

2n.

Using the last equality, the Garding inequality (6.3.35) and writing

an(vi+θn , vi+1

n ) = θan(vi+1n , vi+1

n )+(1− θ)an(vin, v

in)+(1− θ)an(vi

n, vi+1n −vi

n)

we find that

1

2δt

(

|vi+1n − ψn|

2n − |vi

n − ψn|2n + |vi+1

n − vin|

2n

)

+ α(

θ‖vi+1n ‖2

n + (1 − θ)‖vin‖

2n

)

≤ (f i+θn , ψn − vi+1

n )n + an(vi+θn , ψn) − (1 − θ)an(vi

n, vi+1n − vi

n)

+ ρθ|vi+1n |2n + ρ(1 − θ)|vi

n|2n. (6.4.30)

To estimate the first three terms on the RHS, we note that

|(f i+θn , ψn − vi+1

n )n| ≤ |f i+θn |2n + 1

2 |vin|

2n + 1

2 |ψn|2n,

|an(vi+θn , ψn)| ≤ θ|an(vi+1

n , ψn)| + (1 − θ)|an(vin, ψn)|n

≤ cθ‖vi+1n ‖n‖ψn‖n + c(1 − θ)‖vi

n‖n‖ψn‖n

≤ 12θα‖v

i+1n ‖2

n + 14(1 − θ)α‖vi

n‖2n + (1 − 1

2θ)c2

α‖ψn‖

2n,

|an(vin, v

i+1n − vi

n)| ≤ c‖vin‖n‖v

i+1n − vi

n‖2n

≤ 14α‖v

in‖

2n +

c2

α‖vi+1

n − vin‖

2n

≤ 14α‖v

in‖

2n +

c2s(n)2

α|vi+1

n − vin|

2n

where the second and third estimates use relation (6.4.28) variously with ε = cα

and ε = 2cα . Incorporating these estimates in (6.4.30) we have

|vi+1n − ψn|

2n − |vi

n − ψn|2n + (1 − β)|vi+1

n − vin|

2n

+ αδt(

θ‖vi+1n ‖2

n + (1 − θ)‖vin‖

2n

)

≤ δt(

2|f i+θn |2n + |ψn|

2n + (2 − θ)

c2

α‖ψn‖

2n

+ (2ρθ + 1)|vi+1n |2n + 2ρ(1 − θ)|vi

n|2n

)

(6.4.31)


where we have used β from the stability assumption (6.4.20).

Now summing (6.4.31) from i = 0, . . . , k − 1 and noting v0n = ψn, we have

|vkn − ψn|

2n + β

k−1∑

i=0

|vi+1n − vi

n|2n + αδt

k−1∑

i=0

(

θ‖vi+1n ‖2

n + (1 − θ)‖vin‖

2n

)

≤ 2δt

(

k−1∑

i=0

|f i+θn |2n + (1

2 + ρθ)

k−1∑

i=0

|vi+1n |2n + ρ(1 − θ)

k−1∑

i=0

|vin|

2n

+ k(1 − 12θ)

c2

α‖ψn‖

2n + 1

2k|ψn|2n

)

. (6.4.32)

Using the relation 12 |a|

2−|b|2 ≤ |a−b|2 for the first term on the LHS, collecting

the |vin| terms and rearranging we have

|vkn|

2n + 2β

k−1∑

i=0

|vi+1n − vi

n|2n + 2αδt

k−1∑

i=0

(

θ‖vi+1n ‖2

n + (1 − θ)‖vin‖

2n

)

≤ 2(1 + kδt)|ψn|2n + 4δt

k−1∑

i=0

|f i+θn |2n + 2δt(1 + 2ρ)

k∑

i=0

|vin|

2n

+ 2kδt(2 − θ)c2

α‖ψn‖

2n. (6.4.33)

Now we use Lemma 6.4.8 with the first term on the LHS. Letting

c0 = supn

2|ψn|2n

c1 = 2(1 + 2ρ)

c2 = 1

b(t) = supn

4|f(t)|2n + 2|ψn|2n + 2(2 − θ)

c2

α‖ψn‖

2n

we have

lim sup |vkn|

2n ≤ c0e

c1t + c2

∫ t

0ec1(t−s)b(s)ds,

and therefore boundedness of vkn in the | · |n norm for 0 ≤ kδt ≤ T and for n and δt

satisfying the stability condition (6.4.20). This implies a uniform stability estimate

|vkn|

2n ≤ c0e

c1t + c2

∫ t

0ec1(t−s)b(s)ds+ c3


for some c3 > 0.

We now note that this leads to a uniform asymptotic bound on the RHS of

(6.4.33), and thus the other terms on the LHS of (6.4.33) are also uniformly bounded.

6.4.3 Convergence

Theorem 6.4.2 Suppose that (6.2.19) has a unique solution, the stability condition

(6.4.20) holds and that v(n,δt) are the solutions to the discretised systems (6.3.39).

Then

pnv(n,δt) → σv (6.4.34)

as n→ ∞, δt→ 0 provided that (1 − θ)δts(n)2 → 0.

Proof. We refer the reader to Chapter 6, Section 3 of Glowinski et al. [35] for the

details, with the stability of the relevant quantities being ensured through Theorem

6.4.1.

6.5 Approximating the bilinear form

In Example 6.3.4 we suggested using the methods presented in Chapter 4 for ap-

proximation of the bilinear form on an irregular grid. We now investigate properties

of these methods and in particular the manner in which the conditions presented in

Section 6.3.2 are satisfied for these approximations.

6.5.1 Local consistency method

The local consistency method presented in Chapter 4 uses considerations proposed

in Kushner and Dupuis [46] to construct the operator approximations An. Namely,

to construct An on a grid X = x1, . . . , xn one solves for each row i of An,

6.5. Approximating the bilinear form 155

relating to point xi, the feasibility problem

∑

j 6=i

(xj − xi)(xj − xi)′ai,j = Σ (6.5.1)

∑

j 6=i

(xj − xi)ai,j = µ (6.5.2)

ai,j ≥ 0 (6.5.3)

for the closest possible set of neighbouring points1. One then sets ai,i = −∑n

j 6=i ai,j

in order to maintain zero row sums in An.

One may arrive at the conditions (6.5.1) through considering the Taylor series

expansions of the value function

v(xi + δx0) = v(xi) +d∑

j=1

δx0,j∂v

∂xj(xi) +

1

2

d∑

j,k=1

δx0,jδx0,k∂2v

∂xj∂xk(xi)

+O

d∑

j,k,l=1

|δx0,jδx0,kδx0,l|

...

v(xi + δxη) = v(xi) +

d∑

j=1

δxη,j∂v

∂xj(xi) +

1

2

d∑

j,k=1

δxη,jδxη,k∂2v

∂xj∂xk(xi)

+O

d∑

j,k,l=1

|δxη,jδxη,kδxη,l|

around a feasible xi and for neighbouring points x1, . . . , xη and where δxk,j is the

jth component of xk − xi and η = d(d + 3)/2 is the number of nonzero entries

in each feasible row of An. We assume here that δx0 = 0, so that the closest

neighbour of x is the point x itself and the first row above is reduced to a trivial

statement.

Using the weighted norms of Example 6.3.4, the discrete Garding inequality is

1One may consider many measures of the “closeness” of a set of points; in Chapter 4, linear

weightings were considered, and linear programming was used to find the weights.


equivalent to the matrix

ρI − Θ−1/2

(

12(ΘA+A′Θ) + αΘ + α

d∑

i=1

δ′iΘδi

)

Θ−1/2 (6.5.4)

having nonnegative eigenvalues. One may use for example the first derivative ap-

proximations suggested in (6.3.20) for the δi. It suffices to check that the matrix

Θ−1/2

(

12(ΘA+A′Θ) + α

d∑

i=1

δ′iΘδi

)

Θ−1/2 (6.5.5)

has eigenvalues bounded above, as n → ∞ and for some α > 0. The Garding

inequality is then satisfied where ρ is sufficiently large such that ρ − α is at least

equal to this eigenvalue bound.

6.5.2 Experiments

Due to the difficulty in analysing the maximum eigenvalues of (6.5.5) analytically,

we consider experimental evidence for their behaviour. We also conduct experi-

ments to directly investigate stability of the quantities on the LHS of (6.4.33).

The eigenvalue investigation is conducted in dimensions 1–8, and involves

computing the maximum eigenvalue of (6.5.5) numerically using the Matlab sparse

matrix analysis routines. The experiments involved the following steps:

1. constructing the grid X ,

2. constructing the infinitesimal generator A and first derivative matrices δi,

3. constructing the weights matrix Θ,

4. computing the maximum eigenvalue.

This procedure was repeated for various grids, and for various specifications of the

parameters α and µ. In all cases the dynamics involve zero drift and unit covariance

per unit time.

The weights matrix Θ is calculated according to (6.3.18), where the entries are

estimated using quasi-Monte Carlo integration as follows. Let f be an appropriate

importance sampling density (we used the standard normal density in d dimen-

sions), and let Xj , j = 1, . . . , N , be quasi-Monte Carlo points corresponding to f .


Now let Ni be the number of sample points in the Voronoi cell containing the grid

point xi, and denote by Xi,j , j = 1, . . . , Ni, the jth point in this cell. Then

θi =

∫

CX (i)e−µ|x|dx

=

∫

CX (i)

e−µ|x|

f(x)f(x)dx

'1

Ni

Ni∑

j=1

e−µ|Xi,j |

f(Xi,j)

Ni

N

=1

N

Ni∑

j=1

e−µ|Xi,j |

f(Xi,j).

In this way, the θi may all be calculated from a single quasi-Monte Carlo sampling.

We found that the parameter µ made little difference to the qualitative be-

haviour of the eigenvalues, although the results were generally shifted vertically. It

was also found that α = 0.01 was sufficiently small to allow a stable behaviour of

the eigenvalues in most cases.

The results are plotted in Figures 6.5.1–6.5.10. Since the main question in-

volves the boundedness of the eigenvalues with grid size, the maximum eigenvalue

is plotted against the grid size.

In Figure 6.5.1 results are shown for grids constructed from inverse normal

transformations of regular and Sobol’ grids in one dimension. One sees that the

maximum eigenvalues of the inverse normal regular grids show a monotone sub-

log behaviour, while the normal Sobol’ grids show an oscillating behaviour which

seems to increase at a slightly faster rate than the inverse normal regular grid, at

least on average.

Two types of inverse normal regular grid are considered, where the regular

grids are denoted type 1 and type 2. A regular grid of type 1 with n points contains

a point at the center of each interval [k/n, (k + 1)/n] for k = 0, . . . , n − 1, thus

containing the points 1/2n, 3/2n, . . . , (2n − 1)/2n. A regular grid of type 1 with

n points contains the points 1/(n+ 1), . . . , n/(n+ 1). It should be noted that the

Sobol’ grid coincides with the regular grid type 2 for grid sizes n = 2m − 1 where

m is an integer. The Sobol’ grid never coincides with the regular grid of type 1 for

grid sizes n > 1.


0 500 1000 1500 2000 2500 3000 3500 4000 45000.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44m

ax(λ

)

Regular 1Sobol’Regular 2

(a) Natural scale

102

103

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

n

max

(λ)

Regular 1Sobol’Regular 2

(b) Log scale

Figure 6.5.1: Maximum eigenvalues of (6.5.5) for inverse normal regular and

Sobol’ grids in one dimension plotted against grid size n and for µ = 2, α = 0.01.

The regular grid type 1 of size n contains points 1/2n, 3/2n, . . . , (2n−1)/2n, and

the regular grid type 2 of size n contains points 1/(n+ 1), . . . , n/(n+ 1).

In two dimensions we present results for inverse normal regular grids in Fig-

ure 6.5.2 and normal Sobol’ and low distortion grids in Figure 6.5.3. The max-

imum eigenvalue for the inverse normal regular grids shows a monotone sub-log

behaviour to start with, similar to that of the one-dimensional inverse normal regu-

lar grids. At about n = 1000 however, the maximum eigenvalue drops and starts to

behave in a slightly less regular fashion. This behaviour is puzzling, and probably

has to do with the configuration of neighbours used for the infinitesimal generator.

For dimensions 2–8 we consider the use of normal Sobol’ and low distortion

grids. Sobol’ grids are constructed using Sobol’ sequences, which are examples

of low discrepancy sequences as described by Niederreiter [58]. Low distortion

grids are a separate class of point sets which aim to minimise a distortion function

related to the relevant density, in this case the standard normal. These point sets are

usually generated using stochastic descent algorithms, as described by Pages [60];

we used 105 iterations and a step size of k−0.4 to generate the grids (where k is

the iteration number), along with some multiple sampling and quasi-Monte Carlo

adaptations.

The generator matrix is constructed as detailed above, where we attempt to

satisfy the local consistency conditions using 5η nearest neighbours.

The results for normal Sobol’ and low distortion grids in 2–8 dimensions are


0 500 1000 1500 2000 2500 3000 3500 4000 4500−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

n

max

(λ)

(a) Natural scale

102

103

−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

n

max

(λ)

(b) Log scale

Figure 6.5.2: Maximum eigenvalues of (6.5.5) for inverse normal regular grids in

two dimensions plotted against grid size n with µ = 2, α = 0.01.

presented in Figures 6.5.3–6.5.9. The low distortion plots, also shown together

in Figure 6.5.10, show a largely stable monotone and convex behaviour, with the

exception of the two-dimensional case which does not offer evidence of stability.

This may be due to the stochastic nature of the grid generation, which may be less

efficient at smoothing the points in two dimensions.

We also observe less regular behaviour in higher dimensions, which may be

due to the number of points being very low compared to the dimension. This has

the effect that the local consistency conditions may not be satisfied for a large

number of points in the grid, leading to greater asymmetry and a larger number of

zero entries in the generator matrix.

The results for normal Sobol’ grids in Figures 6.5.3–6.5.9 are not conclusive

regarding stability. We tend to see a stable minimum trend with occasional large

deviations which may persist for a number of successive grids. The latter behaviour

is not unexpected for the normal Sobol’ grids since they are related in that each grid

is a superset of the preceding ones. We conjecture that the large eigenvalues are a

result of local roughness in the grids which persists as a result of this relationship.

In addition to the eigenvalue investigations we now provide a direct investi-

gation into the behaviour of the stability quantities on the LHS of (6.4.33). The

experiments conducted involved pricing a geometric average option on one, two

and five assets. The parameters used were identical to those used in Chapter 4, and

we used µ = 2 in the weighting matrix. The stability quantities investigated were


0 500 1000 1500 2000 2500 3000 3500 4000 45000

20

40

60

80

100

120

140

160

180

n

max

(λ)

(a) Sobol’

0 1000 2000 3000 4000 50000

2

4

6

8

10

12

14

16

18

n

max

(λ)

(b) Low distortion

Figure 6.5.3: Maximum eigenvalues of (6.5.5) for normal Sobol’ and low distortion

grids in two dimensions plotted against grid size n with µ = 2, α = 0.01.

0 500 1000 1500 2000 2500 3000 3500 4000 4500−50

0

50

100

150

200

250

n

max

(λ)

(a) Sobol’

0 500 1000 1500 2000 2500 3000 3500 4000 4500−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

n

max

(λ)

(b) Low distortion


grids in three dimensions plotted against grid size n with µ = 2, α = 0.01.


0 500 1000 1500 2000 2500 3000 3500 4000 4500−2

0

2

4

6

8

10

12

14

16

18

n

max

(λ)

(a) Sobol’

0 500 1000 1500 2000 2500 3000 3500 4000 4500−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

n

max

(λ)

(b) Low distortion


grids in four dimensions plotted against grid size n with µ = 2, α = 0.01.

0 500 1000 1500 2000 2500 3000 3500 4000 4500−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

n

max

(λ)

(a) Sobol’

0 500 1000 1500 2000 2500 3000 3500 4000 4500−1.3

−1.2

−1.1

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

n

max

(λ)

(b) Low distortion


grids in five dimensions plotted against grid size n with µ = 2, α = 0.01.


0 500 1000 1500 2000 2500 3000 3500 4000 4500

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

n

max

(λ)

(a) Sobol’

0 500 1000 1500 2000 2500 3000 3500 4000 4500−1.6

−1.5

−1.4

−1.3

−1.2

−1.1

−1

−0.9

n

max

(λ)

(b) Low distortion


grids in six dimensions plotted against grid size n with µ = 2, α = 0.01.

0 500 1000 1500 2000 2500 3000 3500 4000 4500−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

n

max

(λ)

(a) Sobol’

0 500 1000 1500 2000 2500 3000 3500 4000 4500

−1.8

−1.7

−1.6

−1.5

−1.4

−1.3

−1.2

−1.1

−1

−0.9

−0.8

n

max

(λ)

(b) Low distortion


grids in seven dimensions plotted against grid size n with µ = 2, α = 0.01.


0 500 1000 1500 2000 2500 3000 3500 4000 4500−2.2

−2.1

−2

−1.9

−1.8

−1.7

−1.6

−1.5

−1.4

−1.3

−1.2

n

max

(λ)

(a) Sobol’

0 500 1000 1500 2000 2500 3000 3500 4000 4500−1.9

−1.85

−1.8

−1.75

−1.7

−1.65

n

max

(λ)

(b) Low distortion


grids in eight dimensions plotted against grid size n with µ = 2, α = 0.01.

0 500 1000 1500 2000 2500 3000 3500 4000 4500−2

0

2

4

6

8

10

12

14

16

n

max

(λ)

Dimension 2Dimension 3Dimension 4Dimension 5Dimension 6Dimension 7Dimension 8

Figure 6.5.10: Maximum eigenvalues of (6.5.5) for low distortion grids in 2–8

dimensions plotted against grid size n with µ = 2, α = 0.01.


0 500 1000 1500 2000 25000

50

100

150

200

250

300

350

n

max

i |vni | n2 1 Dimension

2 Dimensions5 Dimensions

Figure 6.5.11: Plot of the first stability quantity maxi |vin|

2n versus grid size for

normal Sobol’ grids in dimensions 1, 2 and 5. We used δt = 0.1 and µ = 2.

1. maxi |vin|

2n,

2.∑

i |vi+1n − vi

n|2n,

3. δt∑

i

(

θ‖vi+1n ‖2

n + (1 − θ)‖vin‖

2n

)

.

Figures 6.5.11–6.5.13 show the behaviour of the stability parameters plotted

against grid size. In addition to the information displayed, we found that the time

step had negligible effect on the first and third parameters, and only affected the

level for the second parameter as shown in Figure 6.5.12. Again, altering the

weighting parameter µ only affected the level of the results, and not the qualita-

tive behaviour.

The graphs do not show any particularly unstable behaviour, except for the

third parameter for low grid sizes in five dimensions. This does not seem to be a

cause for concern since the grid sizes concerned are very low for a five-dimensional

problem, and the quantity seems to be stable for larger grid sizes.

6.6 Conclusions

In this chapter we have used the variational inequality framework of Glowinski et

al. [35] to prove convergence of numerical schemes for solving high-dimensional

optimal stopping problems, including the American option problem. The proof

may be applied in particular to the schemes suggested in Chapters 2–4 (also pub-

lished as [8, 7, 9, 10]).


0 500 1000 1500 2000 25000

0.5

1

1.5

2

2.5

n

Σ i |vni+

1 −v ni | n2 1 Dimension


(a) δt = 0.10

0 500 1000 1500 2000 25000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Σ i |vni+

1 −v ni | n2 1 Dimension


(b) δt = 0.02

Figure 6.5.12: Plot of the second stability quantity∑

i |vi+1n −vi

n|2n versus grid size

for normal Sobol’ grids in dimensions 1, 2 and 5 and two different time steps. We

used µ = 2.

0 500 1000 1500 2000 25000

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

n

δ t Σ

i ( θ

||vni+

1 || n2 + (

1−θ)

||vni || n2 )

1 Dimension2 Dimensions5 Dimensions

(a) Full plot

500 1000 1500 2000 2500

0

100

200

300

400

500

600

700

n

δ t Σ

i ( θ

||vni+

1 || n2 + (

1−θ)

||vni || n2 )

1 Dimension2 Dimensions5 Dimensions

(b) Zoomed

Figure 6.5.13: Plot of the third stability quantity δt∑

i

(

θ‖vi+1n ‖2

n +(1 − θ)‖vin‖

2n

)

versus grid size for normal Sobol’ grids in dimensions 1, 2 and 5. The right hand

plot is zoomed to better show the behaviour for dimensions 1, 2 and for larger grid

sizes in dimension 5. We used δt = 0.1 and µ = 2.


The approach is similar to that of Jaillet et al. [42] and Zhang [76], but in con-

trast to these authors we consider a high-dimensional setting in which an irregular

grid is used. The use of such a grid allows one to use a tractable number of points in

the discretisation, as opposed to regular grid schemes where the number of points

grows exponentially with dimension. For example, one can compute approximate

solutions to ten dimensional problems using an irregular grid, whereas for a regular

grid this would be nearly impossible using current computer technology.

The Glowinski et al. [35] framework is sufficient for our problem, except that

we relax the coercivity assumption to a Garding inequality assumption; this af-

fects the proof of stability in their framework. Zhang [76] also assumes a Garding

inequality for proving convergence of a numerical method for pricing American

options on a one-dimensional jump-diffusion process. Our proof rather treats the

multidimensional case, and uses a discrete version of Gronwall’s inequality to

demonstrate stability. The latter provides a clearer understanding of the stability

conditions.

A number of sufficient conditions are stipulated in the convergence proof which

are difficult to check for the suggested schemes. Instead we check certain condi-

tions experimentally for the numerical scheme of Chapter 4 for various specifica-

tions of the relevant parameters.

We find that the maximum eigenvalues related to the matrix (6.5.5) for inverse

normal regular grids in one dimension exhibit a monotone increasing but sub-log

behaviour. The maximum eigenvalues for normal Sobol’ grids in one dimension

show an oscillating behaviour whose mean appears to behave in a similar fashion to

the regular grid case. For the two-dimensional case we observe unusual behaviour

both for the regular and low distortion grids; further investigations are required

here.

In higher dimensions the maximum eigenvalue behaviour is less regular, al-

though for low distortion grids in dimensions 3–8 we observe a mostly monotone

increasing convex dependence on the grid size with some slight aberrations. For the

normal Sobol’ grids it is more difficult to draw conclusions since a lot of variation

is observed in the maximum eigenvalue behaviour.

Finally, a direct investigation of the stability quantities on the LHS of (6.4.33)

does not indicate unbounded behaviour of these quantities with increasing grid size

for dimensions 1, 2 and 5.

Chapter 7

Conclusions and Future Research

This thesis has presented several new numerical methods aimed at the pricing of

high-dimensional American options, and more generally aimed at the efficient so-

lution of high-dimensional optimal stopping problems. The numerical methods

presented are based on irregular grid discretisations of the state space and most

have been shown to be effective for problems of up to ten dimensions.

Chapters 2–4 present methods which use a discrete space Markov chain ap-

proximation to create a related but tractable optimal stopping problem. One may

also see these methods as providing a numerical approach for solving the associ-

ated high-dimensional PDE problems. The experiments presented in these chap-

ters show that the computed solutions are very accurate, as compared to available

benchmarks, with the application of a simple control variate. Chapter 6 shows how

one may prove the convergence of such schemes using the variational inequality

framework developed by Glowinski et al. [35].

Chapter 5 presents a scheme based on value iteration, and using a different ir-

regular grid at each time step. The method is found to work well when an inner

control variate is applied, that is, when a suitable control variate is applied at each

time step. The method is tested in up to ten dimensions and produces results con-

sistent with other authors for the prices of Bermudan swaptions in a LIBOR market

model setting. Surprisingly, the results are quite accurate even when only 100 grid

points are used, and do not improve noticeably when up to 500 points are used.

The key ingredients of the methods presented are the use of randomisation and

the absence of parametric functional approximation methods. The use of random-

167

168 Chapter 7. Conclusions and Future Research

isation includes both Monte Carlo and quasi-Monte Carlo methods, which have

been used with great effect for finding numerical solutions to high-dimensional in-

tegration problems. This thesis shows how one may extend these methods to find

numerical solutions for high-dimensional optimal stopping problems. The absence

of parametric functional approximation sets the methods in this thesis apart from

other methods used for solving American option pricing problems. Methods such

as those suggested by Longstaff and Schwartz [50] and Tsitsiklis and Van Roy [71]

require a clever choice to be made in the selection of basis functions for functional

approximation; the methods presented in this thesis may thus be preferable when

it is not possible to make such a choice.

One of the most difficult aspects of constructing numerical methods based on

irregular grid discretisation turns out to be the stability analysis. One would nat-

urally like to construct methods which are stable, but the connection between the

specification of the method and its stability is often tenuous mathematically. We

have seen two methods in this thesis where stability could be guaranteed on a con-

stant irregular grid, namely those of Chapters 2 and 4. Stability was guaranteed in

the former because the infinitesimal generator was constructed as a root of a diag-

onalisable matrix with eigenvalues in the unit interval, and in the latter through the

use of linear programming in combination with an application of the Gershgorin

disk theorem. We saw in Chapter 3 however that, even though the method presented

is defined in a seemingly consistent manner, one may lack a tractable mathematical

approach for analysing the stability; this makes such a method difficult to apply in

practise.

A key question which remains open is whether the curse of dimensionality can

be beaten for problems of the type considered in this thesis. Evidence is provided

in Chapter 4 that the method used in that chapter suffers from the curse of di-

mensionality in that, in order to obtain an approximation with a certain accuracy,

one requires an exponentially increasing amount of computational work with di-

mension. Rust [65] provides evidence that certain optimal control problems admit

randomised numerical solution methods that do not suffer the curse of dimension-

ality. Such problems have a discrete action space, which is certainly the case for

American options, but also require a compact state space. One may reformulate the

American option problem on a compact state space, but it seems that performing

such a transformation would violate the conditions specified for the dynamics in

169

[65].

In Chapter 5 we saw that the local quadratic interpolation, or moving least

squares method, is very slow in a high-dimensional setting. Recent work by Maz’ya

and Schmidt [53] and Fasshauer [29] suggest that this interpolation may be done

much faster, with a small error as penalty, using so-called matrix-free methods to

avoid the matrix inversion required in the interpolation. Such methods, if success-

ful in a high-dimensional setting, would be useful to make the interpolation method

viable for the type of problem considered in this thesis.

A simple question which remains open is whether one can in practise use an

irregular grid to approximate types of high-dimensional stochastic process other

than the ones considered in this thesis. Given their recent popularity, it would be

of great interest to know for example whether multidimensional Levy processes

could be approximated consistently and efficiently in an irregular grid framework.

Related work by Matache et al. [51, 52] shows how one can efficiently price both

European and American options based on Levy processes using a regular grid dis-

cretisation in one dimension.

The convergence analysis presented in Chapter 6 builds on the variational in-

equality framework introduced by Glowinski et al. [35] to provide a proof of con-

vergence for the type of irregular grid methods introduced in Chapters 2–4. Suf-

ficient conditions are provided for convergence of such methods; however, given

the mathematical complexity of the methods, it turns out to be difficult to for-

mulate conditions which can be checked analytically. We check certain condi-

tions experimentally for the method presented in Chapter 4. It is of great interest

to make stronger statements, including the specification of more easily verifiable

conditions, regarding convergence of the proposed methods. Another approach

that may be considered for constructing a proof of convergence is that of Kush-

ner and Dupuis [46], who use an approximating Markov chain framework to prove

convergence of numerical methods for stochastic optimal control problems.

A natural extension of the methods presented would be to the solution of op-

timal control problems. In related work, Munos and Moore [55, 56, 57] present

adaptive discretisation methods for solving deterministic optimal control problems.

Such methods use irregular discretisations for the purpose of obtaining more ac-

curate representations of the problem in regions of the state space which are more

turbulent, or likely to have a more significant effect on the solution. Though not

170 Chapter 7. Conclusions and Future Research

concentrating in particular on high-dimensional problems, their adaptive methods

include the use of sparse and low discrepancy grids and they do report being able

to solve stochastic optimal control problems in six dimensions; this is not ex-

panded on in the above papers however. Their work demonstrates that irregular

grid methodologies are applicable in deterministic, and possibly stochastic, opti-

mal control problems. The results contained in this thesis concur, at least for the

special case of optimal stopping problems.

Appendix A

Software

The software used for conducting experiments in this thesis was entirely Matlab-

based. Matlab versions 5.3, 6.1 and 6.5 were used at various stages, and use was

made both of the scripting capabilities of Matlab and the ability to compile parts

of the code as mex files, using the C language.

The linear programming problems in Chapter 4 were solved using Tomlab/MI-

NOS v4.1, which contains an implementation of the MINOS optimisation library

developed at Stanford. These routines were found to give faster and more accurate

results than those included in the Matlab optimisation toolbox.

The fast nearest neighbour searching required in Chapters 3–6 was carried out

using the TSTOOL package developed at DPI Gottingen by Christian Merkwirth,

Ulrich Parlitz, Immo Wedekind and Werner Lauterborn. This package is released

under the GNU General Public License, and available from the DPI Gottingen web

site.

Parallel computing was achieved in Matlab using the Parmatlab package de-

veloped by Lucio Andrade, and available online through the Matlab Central File

Exchange. This package also makes use of the TCP/IP toolbox v1.2.3 by Peter Ry-

desater. The Parmatlab package was extended to function in Matlab 6 with version

2.0.2 of the TCP/IP toolbox, and various extra features were added such as more

detailed output information and a timeout capability.

171

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Samenvatting

Dit proefschrift behandelt het probleem van de waardering van Amerikaanse op-

ties in modellen waarbij de beslissing over al of niet uitoefening afhankelijk is van

drie tot tien toestandsvariabelen. In het onderzoek is ook aandacht besteed aan

het verwante probleem van de waardering van Bermudaanse opties, die slechts op

een beperkt aantal tijdstippen uitgeoefend kunnen worden. Modellen met drie of

meer toestandsvariabelen worden “hoogdimensionaal” genoemd omdat klassieke

discretisatiemethoden, gebaseerd op regelmatige roosters, in deze context lastig

toepasbaar zijn. In dit proefschrift worden discretisatiemethoden toegepast met

onregelmatige roosters. Voordelen van het gebruik van een onregelmatig rooster

zijn onder meer vrijheid in het kiezen van het aantal roosterpunten, en vrijheid in

het plaatsen van de roosterpunten in gebieden die belangrijke invloed hebben op de

te bepalen oplossing. Verder blijkt er een gunstig effect te zijn van onregelmatige

roosters op een typisch probleem dat zich voordoet in hoogdimensionale situaties,

namelijk het toenemend aandeel van randpunten in het totaal aantal roosterpunten.

Een belangrijk nadeel van het gebruik van onregelmatige rooosters is dat het dis-

cretiseren van de differentiaaloperatoren niet meer op een voor de hand liggende

manier kan gebeuren. In het proefschrift worden hiervoor verschillende oplossin-

gen gesuggereerd.

De methoden die in het proefschrift worden ontwikkeld zijn niet alleen van

toepassing op Amerikaanse opties; ze kunnen meer in het algemeen worden ge-

bruikt voor stochastische beslissingsproblemen met binaire keuze. De voornaamste

impuls voor het ontwikkelen van geavanceerde numerieke methoden voor hoogdi-

mensionale problemen komt echter uit de financiele wereld. In de moderne fi-

nanciele markten wordt een toenemend aantal contracten verhandeld waarvan de

waarde kan afhangen van een groot aantal onderliggende variabelen. Bovendien di-

181

182 Samenvatting

enen er niet alleen prijzen, maar ook afdekkings- en uitoefeningsstrategieen bepaald

te worden. Producten die op rentestanden gebaseerd zijn, zoals Bermudaanse

swaptions, vormen een groot deel van zulke contracten; andere voorbeelden zijn

reele opties, die toegepast worden in investeringsanalyses, en multi-asset-opties

waarvan de uitbetaling kan afhangen van bijvoorbeeld een aantal verschillende

aandelen.

De mogelijkheid van vervroegde uitoefening, die zich voordoet zowel bij Ame-

rikaanse als bij Bermudaanse opties, levert op zichzelf nog geen grote problemen

op in een-dimensionale modellen. In hoogdimensionale modellen is de waarder-

ing van opties zonder mogelijkheid van vervroegde uitoefening al evenmin lastig:

zulke problemen kunnen relatief snel opgelost worden met behulp van Monte Carlo

of quasi-Monte Carlo integratiemethoden. Het is de combinatie van hoogdimen-

sionaliteit met de mogelijkheid van vervroegde uitoefening die tot een zware op-

gave leidt voor de numerieke analyse, zowel wat betreft het waarderingsprobleem

als wat betreft het bijbehorende afdekkingsprobleem. Er zijn geen oplosmethoden

bekend die niet aanzienlijke rekenkracht vereisen.

Hoofdstukken 2–4 presenteren methoden die gebruik maken van een benader-

ende Markovketen om een verwant maar toch hanteerbaar beslissingsprobleem te

verkrijgen. Deze methoden kunnen ook beschouwd worden als numerieke be-

naderingen van de overeenkomstige partiele differentiaalvergelijkingen. De ex-

perimenten in deze hoofdstukken tonen aan dat de methoden nauwkeurig zijn, in

vergelijking met bekende methoden, als een geschikte stuurvariabele (control vari-

ate) wordt gebruikt. Hoofdstuk 6 laat zien hoe de convergentie van zulke methoden

bewezen kan worden door middel van de analyse van numerieke methoden voor de

benadering van variationele ongelijkheden die is ontwikkeld door Glowinski, Lions

en Tremolieres [35].

Hoofdstuk 5 presenteert een methode die gebaseerd is op waarde-iteratie, en

die gebruik maakt van verschillende onregelmatige roosters op verschillende tijd-

stippen. Deze methode werkt goed als een inwendige stuurvariabele wordt toege-

past, hetgeen wil zeggen dat voor ieder tijdstip afzonderlijk een geschikte stuur-

variabele wordt gebruikt. De nauwkeurigheid van de methode wordt getoetst in

problemen tot dimensie 10; de resultaten komen overeen met die van andere au-

teurs voor de waardering van Bermudaanse swaptions in een LIBOR-marktmodel.

Opmerkelijk genoeg zijn de resultaten erg nauwkeurig, zelfs als er maar 100 pun-

Samenvatting 183

ten gebruikt worden, en verbeteren ze niet merkbaar wanneer het aantal punten

verhoogd wordt tot 500.

Essentiele ingredienten van de gepresenteerde methoden zijn het gebruik van

randomisatiemethoden, en het vermijden van benaderingen in geparametriseerde

functieruimtes. Onder randomisatiemethoden worden hier zowel Monte Carlo

als quasi-Monte Carlo methoden verstaan. De QMC methoden zijn zeer effec-

tief gebleken voor het bepalen van numerieke oplossingen van hoogdimensionale

integratie problemen. Dit proefschrift toont aan hoe deze methoden uitgebreid kun-

nen worden om numerieke oplossingen te vinden voor problemen die betrekking

hebben op het kiezen van een optimaal tijdstip van beeindiging. De afwezigheid

van benaderingen in geparametriseerde functieruimtes onderscheidt de methoden

in dit proefschrift van andere methoden die gebruikt worden voor de waardering

van Amerikaanse opties. Methoden zoals voorgesteld door Longstaff en Schwartz

[50] en Tsitsiklis en Van Roy [71] vereisen een goede keuze van basisfuncties voor

functiebenaderingen; de methoden in dit proefschrift zijn te prefereren als het lastig

is zo’n keuze te maken.

Een van de moeilijkste aspecten van de constructie van op onregelmatige roost-

ers gebaseerde numerieke methoden blijkt de analyse van de stabiliteit te zijn.

Natuurlijk is het belangrijk om methoden te construeren die stabiel zijn, maar het

verband tussen de specificatie van de methode en de stabiliteit is vaak lastig te

bepalen. In hoofdstuk 3 wordt getoond dat, zelfs als de gepresenteerde methode op

een schijnbaar consistente manier gedefinieerd is, een hanteerbare manier om de

stabiliteit te analyseren daarmee niet gegeven hoeft te zijn. Dit maakt het moeilijk

zo’n methode toe te passen in de praktijk. Het proefschrift geeft twee methoden

waarmee stabiliteit gegarandeerd kan worden op basis van een constant onregel-

matig rooster, namelijk in hoofdstukken 2 en 4. In hoofdstuk 2 wordt stabiliteit

gegarandeerd door de infinitesimale generator te construeren als de wortel van een

diagonaliseerbare matrix met eigenwaarden in de eenheidscirkel. In hoofdstuk 4

wordt stabiliteit verkregen door het gebruik van lineaire programmering in combi-

natie met de cirkelsstelling van Gershgorin.

Summary

This thesis addresses the problem of pricing American options where the decision

on whether or not to exercise depends on between three and ten state variables. The

closely related problem of pricing Bermudan options, where the number of exer-

cise opportunities is finite, is included in the scope of the research. For “large” one

may read “at least three”, since this is the dimension in which classical solution

methods, in particular those based on regular grid discretisations, become cumber-

some. The thesis further focuses on methods which use an irregular grid as a basis

for calculations. Advantages of using an irregular grid are that one has freedom in

choosing the number of grid points, and freedom in placing more points in areas

where the behaviour has a greater effect on the required solution. A further advan-

tage of using an irregular grid in high-dimensional situations is that the number of

boundary points increases less quickly as a proportion of the total number of grid

points. An important disadvantage of using an irregular grid is the lack of obvi-

ous methods for discretising the differential operator. This thesis suggests several

methods for performing this discretisation.

In fact the methods in this thesis are not only applicable to American option

pricing, but can also be used for general optimal stopping problems. The prime

motivation for the study of high dimensional problems comes from the financial

world. In modern financial markets one observes an increasing number of con-

tracts whose values may each depend on a large number of underlying variables.

Moreover it is of interest not only to determine the prices, but also the hedging and

exercise strategies. Interest rate products such as Bermudan swaptions constitute

a large class of such contracts; other examples are real options which are applied

in investment analysis, and multiasset options where the payoff may depend for

example on a number of stocks.

185

186 Summary

The early exercise feature, offered by both American and Bermudan options,

does not present a great challenge for one-dimensional problems. Neither does

the valuation of European options in a high-dimensional setting present a great

challenge; such problems can be solved relatively quickly using Monte Carlo or

quasi-Monte Carlo integration methods. In combining high-dimensionality with

the early exercise feature however, one appears to require considerable computa-

tional resources both for the pricing problem and the associated hedging problem.

Chapters 2–4 present methods which use a discrete-space Markov chain ap-

proximation to create a related but tractable optimal stopping problem. One may

also see these methods as providing a numerical approach for solving the associ-

ated high-dimensional PDE problems. The experiments presented in these chap-

ters show that the computed solutions are very accurate, as compared to available

benchmarks, with the application of a simple control variate. Chapter 6 shows how

one may prove the convergence of such schemes using the variational inequality

framework developed by Glowinski, Lions and Tremolieres [35].

Chapter 5 presents a scheme based on value iteration, and using a different ir-

regular grid at each time step. The method is found to work well when an inner

control variate is applied, that is, when a suitable control variate is applied at each

time step. The method is tested in up to ten dimensions and produces results con-

sistent with other authors for the prices of Bermudan swaptions in a LIBOR market

model setting. Surprisingly, the results are quite accurate even when only 100 grid

points are used, and do not improve noticeably when up to 500 points are used.

The key ingredients of the methods presented are the use of (quasi-)random-

isation and the absence of parametric functional approximation methods. The

use of randomisation includes both Monte Carlo and quasi-Monte Carlo meth-

ods, which have been used with great effect for finding numerical solutions to

high-dimensional integration problems. This thesis shows how one may extend

these methods to find numerical solutions for high-dimensional optimal stopping

problems. The absence of parametric functional approximation sets the methods

in this thesis apart from other methods used for solving American option pricing

problems. Methods such as those suggested by Longstaff and Schwartz [50] and

Tsitsiklis and Van Roy [71] require a clever choice to be made in the selection of

basis functions for functional approximation; the methods presented in this thesis

may thus be preferable when it is not possible to make such a choice.

Summary 187

One of the most difficult aspects of constructing numerical methods based on

irregular grid discretisation turns out to be the stability analysis. One would nat-

urally like to construct methods which are stable, but the connection between the

specification of the method and its stability is often tenuous mathematically. In

Chapter 3 we see that even though method may be defined in a seemingly consistent

manner, one may lack a tractable mathematical approach for analysing the stability.

This makes such a method difficult to apply in practise. The thesis presents two

methods where stability could be guaranteed on a constant irregular grid, namely

those of Chapters 2 and 4. Stability was guaranteed in the former because the

infinitesimal generator was constructed as a root of a diagonalisable matrix with

eigenvalues in the unit interval, and in the latter through the use of linear program-

ming in combination with an application of the Gershgorin disk theorem.

tilburg university irregular grid methods for pricing high

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